errors in numerical solutions of shock physics problems

Errors in Numerical Solutions of Shock

Physics Problems

A Dissertation Presented

by

Yan Yu

to

The Graduate School in Partial Fulfillment of the Requirements for the

Degree of

Doctor of Philosophy

in

Applied Mathematics and Statistics

Stony Brook University

December 2004

Copyright c© byYan Yu2004


The Graduate School

Yan Yu

We, the dissertation committee for the above candidate for the Doctor ofPhilosophy degree, hereby recommend acceptance of this dissertation.

James GlimmAdvisor

Department of Applied Mathematics and Statistics

Xiaolin LiChairman


Yongmin ZhangMember


Roman SamulyakOutside Member

Brookhaven National LaboratoryCenter for Data Intensive Computing

This dissertation is accepted by the Graduate School.

Graduate School

ii

Abstract of the Dissertation

Errors in Numerical Solutions of ShockPhysics Problems

by

Yan Yu

Doctor of Philosophy

in

Applied Mathematics and Statistics


2004

Advisor: James Glimm

We seek error models for shock physics simulations which are robust and

understandable. We propose statistical models of uncertainty and error in

numerical solutions. To represent errors efficiently in shock physics simulations

we propose a composition law. The law allows us to estimate errors in the

solutions of composite problems in terms of the errors from simpler ones. We

formulate and validate this composition law for shock interactions in planar

geometry. We also explore complications introduced by spherical flow in the

analysis of errors in the numerical solutions. We illustrate that idea in a very

simple context.

iii

For shock interactions in spherical geometry, we conduct a detailed anal-

ysis of the errors. One of our goals is to understand the relative magnitude of

the input uncertainty vs. the errors created within the numerical solution. In

more detail, we wish to understand the contribution of each wave interaction

to the errors observed at the end of the simulation.

Key Words: uncertainty quantification, error model, composition law,

Riemann problem.

iv

To my parents and my loving husband

Table of Contents

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . xix

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 FronT ier Front Tracking Algorithm . . . . . . . . . . . . . . 7

1.2.1 Interface propagation . . . . . . . . . . . . . . . . . . . 7

1.2.2 Interior computations . . . . . . . . . . . . . . . . . . . 10

1.3 The Wave Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Dissertation Organization . . . . . . . . . . . . . . . . . . . . 14

2 Error Analysis for Planar Geometry . . . . . . . . . . . . . . 16

2.1 Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 The Statistical Numerical Riemann Problem . . . . . . . . . . 17

2.2.1 The Isolated Shock and Contact Wave . . . . . . . . . 19

2.2.2 Shock Contact Interactions . . . . . . . . . . . . . . . . 23

vi

2.2.3 Shock Crossing Shock Interactions . . . . . . . . . . . . 28

2.2.4 The Contact Reshock Interactions . . . . . . . . . . . . 29

2.3 The Composition Law . . . . . . . . . . . . . . . . . . . . . . 30

2.3.1 A Multipath Integral for a Nonlinear Multiscattering

Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.2 Evaluation of the Multipath Integral . . . . . . . . . . 33

2.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4.1 Errors in Fully Resolved Calculations . . . . . . . . . . 35

2.4.2 Errors in Under Resolved Calculations . . . . . . . . . 38

3 Error Analysis for Spherical Geometry . . . . . . . . . . . . 41

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 The Statistical Numerical Riemann Problem . . . . . . . . . . 43

3.2.1 The Single Propagating Wave . . . . . . . . . . . . . . 44

3.2.2 The Shock Contact Interaction . . . . . . . . . . . . . 47

3.2.3 Shock Reflection at the Origin . . . . . . . . . . . . . . 51

3.2.4 The Contact Reshock Interaction . . . . . . . . . . . . 52

3.3 Composite Shock Interaction Problems . . . . . . . . . . . . . 54

3.4 Error Decomposition . . . . . . . . . . . . . . . . . . . . . . . 55

4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.1 2D Shock Wave Interactions in Perturbed Spherical Geometries 64

5.1.1 Single Mode Perturbed Interface . . . . . . . . . . . . . 65

5.1.2 Chaotic Mixing . . . . . . . . . . . . . . . . . . . . . . 66

vii

6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.1 Complete List of Ten Riemann Problems (Planar Geometry) . 69

6.2 Errors in Resolved Calculations . . . . . . . . . . . . . . . . . 81

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

viii

List of Figures

1.1 Flow chart for the front tracking computation. With the ex-

ception of the i/o and the sweep and communication of interior

points, all solution steps indicated here are specific to the front

tracking algorithm itself. . . . . . . . . . . . . . . . . . . . . . 8

1.2 Schematic diagram illustrating the operation of a wave filter. Left:

computational data (squares) are fit to an error function. The er-

ror function depends on four parameters, a position, a width, and

two asymptotic values. These determine the wave position, width

and height, with subgrid accuracy. Right: a piecewise linear con-

struction is fit to the rarefaction or compression wave data. . . 13

2.1 Left. Space time density contour plots for the multiple wave inter-

action problem studied in this section. Right. Pressure contour

plots for the base case considered here. . . . . . . . . . . . . . . 18

2.2 Left: Type and location of waves as determined by our wave

filter analysis for the base case considered here. Right: Schematic

representation of the waves and the interactions, with labels for

the interactions, taken from the left frame. . . . . . . . . . . . 19

ix

2.3 Ensemble mean shock width and the standard deviation of the

shock width (left frame). The mean width, equal to about 2∆x,

is much larger than the standard deviation, indicating that the

mean width is essentially a deterministic feature of the solution.

Convergence properties of the travelling wave to the steady state

values on each side of the wave (right frame). The straight line in

the right frame is the asymptote to the exponential convergence

rate, with slope 0.01 in units of ∆x. . . . . . . . . . . . . . . . 20

2.4 Ensemble mean contact width for isolated noninteracting waves.

Because the width is entirely grid related, we record width in units

of ∆x and time in units of the number of time steps. The standard

deviations are also plotted, and are the points to the extreme left

in each frame. Left (step down): we observe an increase from 2

cells to 30 over 104 steps and an asymptotic growth rate cct1/3,

where cc ∼ 1 depends on the flow Mach number. The straight

line in the left frame is the asymptote to the contact width, with

slope 3. Right (step up): We observe a bound on the contact

width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Left: ensemble mean shock and contact widths before and after

a shock contact (step up) interaction and standard deviations.

Right: ensemble mean shock and contact position errors as a

function of time, expressed in grid units. Step up case. . . . . . 24

x

2.6 The solution and its errors at the point (x, t) can be obtained

by “adding up” the solution and errors for the waves within the

domain of dependence . . . . . . . . . . . . . . . . . . . . . . . 32

3.1 Left: space time density contour plot for the multiple wave in-

teraction problem studied in this chapter, in spherical geometry.

Right: type and location of waves determined by the wave filter

analysis with labels for the interactions. Here I.R. denotes the

inward moving rarefaction. . . . . . . . . . . . . . . . . . . . . . 43

3.2 Left. Mach number vs. radius for a single inward propagating

shock. Right. The same data plotted on a log-log scale. . . . . . 45

3.3 Left. Mach number vs. radius for an outward moving shock wave

starting at different radii r0. Right. The same data plotted on

a log-log scale; the dashed lines in this plot represent the power

law model (3.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4 Ensemble mean contact width for a single propagating contact.

We record the width in units of ∆x. The standard deviation is

also plotted, as the points to the extreme left. . . . . . . . . . . 47

3.5 Left: ensemble mean inward/outward moving shock and contact

widths after a shock contact interaction. Right: ensemble mean

shock and contact position errors as a function of time, expressed

in grid units. The associated standard deviations are extremely

small, not shown in the plots. In the legend, C. denotes the

contact while I.S. and O.S. are the inward and outward moving

shocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

xi

3.6 Schematic graph showing all six wave interaction contributions

to the errors or uncertainty in the output from a single Riemann

solution, namely the reshock interaction (numbered 3 in the right

frame of Fig. 3.1) of the reflected shock from the origin as it

crosses the contact. The numbers labeling the circles refer to

the Riemann interactions contributing to the error. The numbers

labeling the line segments refer to the different error propagating

paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.7 Pie charts showing the contribution of each wave interaction dia-

gram to the error variance of the wave strength at the output of

interaction 3, for a solution using 500 mesh units. . . . . . . . . 59

3.8 Pie charts showing the contribution of each wave interaction dia-

gram to the error variance of the wave strength at the output of

interaction 3, for a solution using 100 mesh units. . . . . . . . . 60

5.1 Density plot for a spherical implosion simulation with a perturbed

interface (single mode). The grid size is 200 × 200. . . . . . . . 66

5.2 Density plot for a spherical implosion simulation with a chaotic

perturbed interface (multiple modes). The grid size is 200 × 200. 67

6.1 Problem 1: Shock-contact (step up) . . . . . . . . . . . . . . . . 70

6.2 Problem 2: Shock-wall interaction . . . . . . . . . . . . . . . . . 70

6.3 Problem 3: Contact-shock (step down) . . . . . . . . . . . . . . 71

6.4 Problem 4: Rarefaction-wall . . . . . . . . . . . . . . . . . . . . 72

6.5 Problem 5: Contact-rarefaction . . . . . . . . . . . . . . . . . . 73

xii

6.6 Problem 6: Shock-shock overtake (two waves of the same family) 74

6.7 Because the width is entirely grid related, we record width in

units of ∆x and time in units of the number of time steps. . . . 76

6.8 Problem 7: Compression-wall . . . . . . . . . . . . . . . . . . . 77

6.9 Problem 8: Contact-compression . . . . . . . . . . . . . . . . . . 79

6.10 Problem 9: Rarefaction-wall . . . . . . . . . . . . . . . . . . . . 80

6.11 Problem 10: Contact-rarefaction . . . . . . . . . . . . . . . . . . 81

xiii

List of Tables

2.1 Parameters that define the base case for the shock contact inter-

action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 The SNRP shock contact (step up) interaction. Expansion coeffi-

cients for output wave strengths, wave widths and wave position

errors (linear model) for input variation ±10%. Here the base

case input contact wave width is zero. . . . . . . . . . . . . . . . 26

2.3 The SNRP defined by the crossing of two shocks. Expansion

coefficients for output wave strengths, widths and position errors

(linear model) for input variation ±10%. . . . . . . . . . . . . . 28

2.4 The SNRP shock contact (step down) interaction. Expansion

coefficients for output wave strengths, widths and position errors

(linear model) for input variation ±10%. . . . . . . . . . . . . . 30

2.5 Predicted and simulated errors for output wave strengths, wave

widths and wave positions, Interaction 1. . . . . . . . . . . . . . 36





xiv

2.8 Case 1. The contact-shock interaction (step up). Errors for out-

put wave strengths, wave widths and wave position. Comparison

of under resolved simulation and prediction. . . . . . . . . . . . 39

2.9 Case 2. The shock crossing equal shock (wave reflection) inter-

action. Errors for output wave strengths, wave width and wave

position. Comparison of under resolved simulation and prediction. 39

2.10 Case 3. The contact-shock interaction (step down). Errors for

output wave strengths, wave width and wave position. Compari-

son of under resolved simulation and prediction. . . . . . . . . . 40

3.1 Comparison of the exponents from the approximate and the exact

similarity solutions for an inward propagating spherical shock wave. 44

3.2 The SNRP shock contact interaction. Expansion coefficients for

output wave strengths, wave strength errors, wave width errors

and wave position errors (linear model) for the initial shock con-

tact interaction. Here the base case input contact wave width

is zero. The final columns refer to difference between the linear

model (2.2) and the exact quantity. The errors in rows 4-12 refer

to the difference between the numerical solution on 100 cells and

the exact solution using 2000 cells. . . . . . . . . . . . . . . . . 48

3.3 The SNRP defined by the shock reflection at the origin. Expan-

sion coefficients for output wave strengths, wave strength errors,

wave width errors and wave position errors (linear model) for in-

put variation ±10%. . . . . . . . . . . . . . . . . . . . . . . . . 51

xv

3.4 The SNRP contact reshock interaction. Expansion coefficients for

output wave strengths, wave strength errors, wave width errors

and wave position errors (linear model). . . . . . . . . . . . . . 53


widths and wave positions, output to interaction 3. The inward

rarefaction and contact strengths are expressed dimensionlessly as

Atwood numbers. The outward shock strengths are in the units of

Mach number. The width and position errors are in mesh units.

The wave strength errors are expressed as mean ± 2σ where σ is

the ensemble STD of the error/uncertainty. . . . . . . . . . . . 55

3.6 The contribution of each interaction to the mean value of the total

error in each of three output waves at the output to interaction 3,

for 100 and 500 mesh units. Units are dimensionless and represent

the error expressed as a fraction of the total wave strength. The

last two rows compare the total of the mean error as given by the

model to the directly observed mean error. The columns I.R., C.,

and O. S. are labeled as in Fig. 3.1, Right frame. . . . . . . . . . 57

6.1 Case 4. The SNRP defined by the crossing of two rarefactions.

Expansion coefficients for output wave strengths (linear model)

for input variation ±10%. . . . . . . . . . . . . . . . . . . . . . 72

6.2 Case 5. The SNRP defined by the contact rarefaction interaction.



xvi

6.3 Case 6. The SNRP defined by the shock shock overtake (two

waves of the same family). Expansion coefficients for output wave

strengths (linear model) for input variation ±10%. . . . . . . . . 77

6.4 Case 7. The SNRP defined by the crossing of two compressions.



6.5 Case 8. The SNRP defined by the contact compression inter-

action. Expansion coefficients for output wave strengths (linear

model) for input varation ±10%. . . . . . . . . . . . . . . . . . 79

6.6 Case 9. The SNRP defined by the crossing of two rarefactions.



6.7 Case 4. The crossing of two rarefactions. Predicted and simulated

errors for output wave strengths, wave widths and wave positions. 82

6.8 Case 5. The contact rarefaction interaction. Predicted and sim-

ulated errors for output wave strengths, wave widths and wave

positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.9 Case 6. The shock shock overtake. Predicted and simulated errors

for output wave strengths, wave widths and wave positions. . . . 84

6.10 Case 7. The crossing of two compressions. Predicted and sim-


positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

xvii

6.11 Case 8. The contact compression interaction. Predicted and sim-


positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.12 Case 9. The crossing of two rarefactions. Predicted and simulated

errors for output wave strengths, wave widths and wave positions. 85

xviii

Acknowledgements

I would like to express my profound gratitude to my advisor, Professor

James Glimm, for suggesting this important and exciting thesis topic and for

his advice, support and guidance toward my Ph. D. degree. He taught me not

only the way to do scientific research, but also the way to become a professional

scientist. He is my advisor and a lifetime role model for me.

I am also deeply indebted to the support of Professor Xiaolin Li. His

scientific vigor and dedication makes him a great mentor and a good friend.

I would like to thank Drs. John Grove, Kenny Ye, Zhiliang Xu, Roman

Samulyak, Dahai Yu, Yongming Zhang and Ning Zhao, from whom I have

learned many important scientific and mathematical skills. Work has been

more productive and life a little easier thanks to them.

I would also like to thank all my friends for their friendship and encour-

agement during my four year study as a graduate student at Stony Brook.

It has been great to have so many friends who can share my sense of hu-

mor, only a few yards away. In particular, I would like to mention Taewon

Lee, Ming Zhao, Erwin George, Andrea Marchese, Xinfeng Liu, Yoonha Lee,

Srabasti Dutta and Xiaofei Fan. They have shared with me many interesting

and inspiring ideas.

Throughout my academic career, the constant support of my parents and

my husband has always motivated me to strive forward. Their unconditional

love has never been affected by the physical distance between us. My disser-

tation is dedicated to them.

During my years here at Stony Brook I have grown and matured both

personally and professionally. I have had my greatest life experience, as well

as the worst. I am grateful to have gone through it all, and I look forward to

what comes next. I thank all who have helped me through this leg of journey.

xx

Chapter 1

Introduction

We begin Chapter. 1 with a background review of uncertainty quantifi-

cation to numerical simulations and the goal of our research study. In Sec. 1.2

and Sec. 1.3, we introduce the FronT ier Front Tracking algorithm for numer-

ical simulation and the wave filter as the diagnostic tool.

1.1 Background

Our approach to numerical solution errors is motivated by needs of un-

certainty quantification. Specifically the Bayesian likelihood is (up to normal-

ization) a probability, which specifies probability of occurrence of an error of

any given size. Unlike other authors [2, 6, 26, 33] who usually use observational

errors or expert opinions to form the probability model for the likelihood, our

approach is to use solution error models for the likelihood. We provide a

scientific basis for the probabilities associated with numerical solution errors.

The authors are not aware of comparable error analysis studies, but of course

numerical simulation errors have been long studied from different points of

1

view.

An early focus of numerical error modeling was round off errors. For the

hyperbolic systems we study, modern 64-bit processors with double precision

arithmetic appear, as a practical matter, not to be sensitive to this class of

errors, while they are difficult to analyze theoretically. A more common ap-

proach to error analysis in numerical analysis is the study of the asymptotic

behavior of errors under mesh refinement. The method of asymptotic analysis

of numerical solution errors is so old and well established that it is difficult to

cite its origins [38]. This is a useful approach, and one we refer to in the case

of well resolved simulations. However, we want an analysis which is also appli-

cable to the pre-asymptotic case of under resolved simulations as these are so

typical of practical studies of realistic complex physical systems. Moreover, the

coefficients which multiply powers of ∆x in the asymptotic expressions cannot

be determined theoretically. A third main theme in the analysis of errors is

the use of a posteriori error estimators. The method of a posteriori analysis

aims to construct an upper bound on the solution error, either theoretically

validated or based on numerical experiments [1, 4, 36, 37]. This method has

been difficult to apply to nonlinear hyperbolic systems, and in any case does

not answer the questions addressed here. We seek to characterize the error,

not just to bound it. Moreover, a posteriori methods are most fully developed

and justified theoretically for elliptic problems, and have only a partial or pre-

liminary development for the shock interaction problems we consider here. A

fourth approach to errors is to regard them as due to input uncertainties. In

this point of view, uncertainty analysis is a mapping of input random variables

2

to output random variables.

We are primarily interested in errors in a preasymptotic range, so while

these methods provide a theoretical framework which is valuable to our anal-

ysis, they do not provide the detailed estimates which we require. In any case

our use of wave filters to diagnosis error leads to more precise measures of

error than is normally considered in the asymptotic analysis.

In contrast to the first three of the above methods, we analyze the errors

statistically. The statistical models are simple, and in this sense, we analyze

only the central portion of the error statistics, not their tails. We use a Gaus-

sian error model, and thus we identify the mean error and the variance. The

mean error is a systematic error, and it can in principle be used to modify or

“post process” and error-correct the approximate solution. The variance is a

measure of the error variability. One may question the idea that numerical

errors can be modeled statistically or that the errors are variable, when each

simulation is totally deterministic. This philosophical question has an easy

answer: determinism lies in the eye of the beholder. In other words, the mod-

eling of a natural phenomena (tossing a coin, for example) as a probabilistic

or deterministic event depends on the level of detail included or excluded from

the model. Thus we argue only why it is convenient or nearly essential to

omit from the modeling of error information needed to make the error analysis

deterministic. The variable (as opposed to the systematic) part of the error

depends on “accidental” features of the numerical model, such as the sub grid

location of waves and wave interaction points relative to the mesh cell edges.

Clearly a deterministic model requiring such data would be too cumbersome

3

for use in practice, and thus a probabilistic model is preferable. Indeed, the

essentially probabilistic aspect of round off error is well recognized. Given that

the complexity of a statistical model is needed, we found that simple linear

error models were sufficient for our analysis [11]. The simple reason for this

pleasant turn of events is that the error is similar to a perturbation, and nor-

mally a small perturbation. Thus the error of a strongly nonlinear problem

still has a useful linear expression, at least as far as our analysis of the error

has progressed.

In contrast to the fourth approach to uncertainty quantification above as

a mapping of uncertainty from input to output, we allow for errors generated

within the solution processes. Thus we subsume and expand on this point of

view.

Simulation errors typically consist of

• position errors in the location of the travelling wave discontinuities or

sharp solution gradients;

• wave width errors in the numerical vs. the physical width of the travelling

waves;

• solution state errors in the smooth regions bordering the regions with

discontinuities or sharp solution gradients.

Any or all of these errors may arise in the input data to the Riemann problem.

Output errors, however, have two sources. Those arising from inaccuracies

in the solution algorithm are called created errors, while those that can be

4

ascribed directly to input error or uncertainty are called transmitted errors or

transmitted uncertainty.

In recent years, three major ideas have been used to develop an approach

to uncertainty quantification to numerical simulations:

1. A combined approach for forward as well as inverse propagation of un-

certainty [22, 23]. This combined approach is important when the use of

disparate sources of data, including data pertaining to observations of

full system performance, is important.

2. Error models for numerical solution errors [7, 17–19]. For multiscale

problems and complex multiphysics problems, under resolved simula-

tions and accompanying simulation error are frequently unavoidable in

practice.

3. Parameterizations, comprehensibility, and validation of simulation error

models [16]. This step allows testing and validation to occur in somewhat

idealized situations, less complex than the full system simulations, but

still applicable to them.

Our principal concern in this dissertation is to develop a method for

determining solution errors in shock wave interaction problems. Our strategy

takes advantage of the fact that shock problems typically consist of smooth

regions which separated by discontinuities (actually narrow regions with strong

solution gradients). The method consists of three main steps.

5

1. Determine solution error models for a comprehensive set of elementary

wave interactions. These are summarized as a set of input/output rela-

tions for the errors in such interactions;

2. Construct wave filters that decompose a complex flow into approximately

independent components consisting of elementary waves;

3. Formulate a composition law that constructs the total solution error at

any space-time point in terms of errors from repeated elementary inter-

actions.

We validate this method by predicting the errors in the composite (com-

plex) simulation with the errors calculated using the composition law [11]. The

results are derived by a study on errors in the solution of Riemann problems.

The Riemann problem is a simple jump discontinuity between two hydro-

dynamic states in one spatial dimension; its idealized solution is called the

Riemann solution [3]. We study the statistical numerical Riemann problem

(SNRP). In SNRP, incoming shock waves have finite width determined from

numerical as well as physical considerations, if specified numerically. Incoming

rarefaction and compression waves have physical time dependent widths. Also

we note that the numerical algorithm that solves the Riemann problem cre-

ates (as well as propagates) errors. We will show that, to fairly good accuracy,

one can model the errors in the outgoing waves as affine linear, i.e., constant

plus linear (or perhaps bilinear) statistical expressions in the strength of the

incoming waves.

6

1.2 FronT ier Front Tracking Algorithm

Since most of our work is based on numerical simulations using FronT ier

code, we present here a brief introduction to this code and its Front Tracking

algorithm.

FronT ier code is based on front tracking, a numerical method which

regards the fluid interface as an evolving, lower dimensional mesh. The Front

Tracking method initiated by Richtmyer and Morton [39] is designed to pro-

vide piecewise smooth solutions as well as distinguished discontinuities which

separate the smooth solutions in solving the system of hyperbolic partial dif-

ferential equations. Fig. 1.1 is a flow chart taken from [14] describing the front

tracking computation in FronT ier. For more details, see [13–15].

Two tasks need to be accomplished at one time step in the Front Track-

ing method. The first is to dynamically evolve the front. The second is to

calculate the numerical solutions in smooth regions surrounded by the fronts.

Corresponding to these two tasks, the Front Tracking method consists of the

following modules.

1.2.1 Interface propagation

An interface is a collection of geometric objects, such as POINT s, CURV Es,

and SURFACEs, that correspond to zero, one, and two dimensional mani-

folds respectively. The interfaces are represented explicitly as lower dimen-

sional meshes. They divide the computational domain into connected regions.

1. The interface points are first propagated normally. By computing the

7

Figure 1.1: Flow chart for the front tracking computation. With the exceptionof the i/o and the sweep and communication of interior points, all solutionsteps indicated here are specific to the front tracking algorithm itself.

8

solution to the local Riemann problem with initial states being those on

either side of the interface point, a wave speed and hence a new position

for the interface point are predicted. These values are only intermediate,

as they do not account for other wave interactions and they assume

constant states on both sides of the discontinuity. The Riemann solution

is usually solved by Newton or secant iteration methods.

2. The method of characteristics is used to linearly trace back to states at

the previous time step. For tracked contact discontinuities (where den-

sity and energy are discontinuous, but pressure and normal velocity are

continuous) as we study in this document, there is one characteristic on

either side of the discontinuity. The states at the previous time step (at

the foot of each characteristic) are computed by interpolating between

the original states on either side of the interface, and those one mesh

unit away in the normal direction.

3. These interpolated states with the Rankine-Hugoniot conditions deter-

mine a solution to the discretized characteristic equations, which pro-

vides an update to the left and right states of the predicted new position.

4. With these updated states at new position, we solve a Riemann problem,

and compute an updated wave speed as well as even further updated

states.

5. The front speeds from both Riemann problems (step 1 and 4) are aver-

aged to compute the final front speed, and this speed is used to compute

the final new position of the interface point.

9

6. The interface states are then updated by a tangential sweep, which uses

a chosen interior solver with a stencil centered at the new interface point.

7. Any tangles in the interface are resolved.

8. At user-defined intervals, the interface points are redistributed to reduce

the variance in size and aspect ration of the segments (2D) or triangles

(3D) making up the interface.

1.2.2 Interior computations

A connected region in the domain separated by the interface is repre-

sented by a component. Therefore, each grid node is associated with a specific

component in addition to the state variables. The interior states are updated

by finite difference schemes.

The interior computations are dimensionally split, so that we may use

simpler, one-dimensional schemes. The order of the one-dimensional sweeps

is changed periodically to avoid incurring first order simulation error (for de-

tails, see [35]). Computations near the fluid interface use ghost cells [21] to

avoid crossing the interface, keeping the different fluid computations entirely

separate. Computational experiments have shown the use of ghost cells to be

reasonably conservative [14]. Our choice of interior solver for the work in this

document is the MUSCL scheme.

10

1.3 The Wave Filter

We introduce diagnostic windows that measure the solution state in one

of the constant regions between the waves as well as wave filters that diagnose

the wave type (only regions with a single wave pass through the filter). In doing

so, we first summarize and then extend the results of [32]. The moving window

in the wave filter has an initial width of 5 cells for shock and rarefaction waves

and 11 cells for contacts. The choice of these parameters appears to be suitable

for most higher order Godunov schemes. In this window, a Riemann problem is

solved using the extreme left and right states as input. The Riemann solution

has 3 outgoing waves, whose strengths are assessed dimensionlessly in terms

of density and pressure differences and ratios. According to these strengths,

and a suitable cutoff for the strength, we identify from zero to three of the

waves as strong, and only the case of a single strong wave is analyzed further.

If adjacent or overlapping windows show a single identical wave, the windows

are merged, so that the full width of the wave will be brought into a single

window. This merging of adjacent or overlapping windows of increasing size

continues recursively until the same wave type fails to show up in the adjacent

windows.

Wave profiles are reconstructed using fitting functions of the form:

ρ(x, t) = ρ− +ρ− − ρ+

2

(f(

x − xc(t)√2σ

) + 1

)(1.1)

where ρ± refer to the asymptotic values for density ahead or behind the wave,

xc(t) is the moving center of the wave, and 2σ is the measure of the wave

11

width. The fitting function f(x) is either the erf function

f(x) = erf(x) =2√π

∫ x

0

e−t2dt (1.2)

for contact or shock waves, or a linear ramp:

f(x) =

−1

x

1

x < −1

−1 < x < 1

1 < x

(1.3)

for rarefactions and compressions.

States identified as within an active region for a single shock or contact

wave by the filter are fit to an error function. The fitting allows determination

of the location of the wave (with subgrid accuracy, up to O(∆x2)), and its

width. See Fig. 1.2, Left. For single rarefaction or compression waves, the

waves are fit to a straight line segment, linear in the characteristic speed

variable. Thus the region of constant and variable states are fit to three

straight line segments, the two extreme ones being constants. See Fig. 1.2,

Right. The width for a shock or contact wave is defined in terms of the

error function fit to the shock or contact wave profile. Let σ be the standard

deviation that enters into the definition of the error function. Then the width

(in units of ∆x) is the distance needed for a 2σ transition (between 2.3% and

97.7%) of the jump in the density (for a contact wave) or in the pressure (for a

shock wave). The width of a rarefaction or compression wave is defined as the

distance between the edges of the central linear piece for its piecewise linear

12

description. The position is defined, with subgrid accuracy, as the position of

the mean value, at a point half way through the jump.

Figure 1.2: Schematic diagram illustrating the operation of a wave filter. Left:computational data (squares) are fit to an error function. The error functiondepends on four parameters, a position, a width, and two asymptotic values.These determine the wave position, width and height, with subgrid accuracy.Right: a piecewise linear construction is fit to the rarefaction or compressionwave data.

The wave filter is the fundamental diagnostic tool that identifies individ-

ual waves, here within the solution of a numerical Riemann problem and in

Sec. 2.3 within the solution of a complex wave interaction problem. We note

immediately a limitation of the methodology, at least as presently developed.

The definition of the wave filters assumes that the individual waves in the

Riemann problem have separated. For sufficiently coarse grids in the wave in-

teraction problem of Sec. 2.3, the waves will enter into new interactions before

clearly separating as they leave an earlier interaction. A second, and related

limitation concerns the relaxation of the left and right states at the edge of a

13

wave to their far field values, an issue studied in Sec. 2.2.1. If a subsequent

interaction occurs before this relaxation is complete, the associated errors will

be “frozen” into the input and output of this later interaction. We will assess

this issue in Sec. 2.3.

1.4 Dissertation Organization

In Chapter. 1, we gave a background review of uncertainty quantification

to numerical simulations and the goal of our research study. We have also

introduced the FronT ier Front Tracking algorithm for numerical simulation

and the wave filter as the diagnostic tool. The rest of this dissertation will be

devoted to analyze the errors in numerical solutions of shock physics problems.

Chapter. 2 present the composition law for complex 1D shock wave in-

teraction problem in planar geometry. The problem is generated by a shock

wave interacting with a contact in the vicinity of a wall. Multiple reflections

between the wall and contact generate a large number of Riemann problems,

that comprise the major features of this problem. The errors can be com-

puted in two ways, directly by statistical analysis of the data and combining

the errors created by and propagated through the individual Riemann prob-

lems using the composition law. The comparison of errors for the complex,

multi-interaction problem, thus determined in two ways, provides validation

for our proposed composition law for errors.

In Chapter. 3, we study the composition law in spherical geometry. We

explore complications introduced by spherical flow in the analysis of errors in

the numerical solutions. We also conduct a detailed analysis of the errors to

14

understand the contribution of each wave interaction to the errors observed at

the end of the simulation.

The conclusion and a discussion of remaining open problems are presented

in Chapter. 4 and 5. Finally, in the appendix, we present the complete set of

Riemann problems (planar geometry) that are studied.

15

Chapter 2

Error Analysis for Planar Geometry

2.1 Problem Setup

We consider, in one spatial dimension, the interaction of a shock wave

with a contact located near a reflecting wall. The base case for the first wave

interaction is defined in Table 2.1. In this table, P represents the pressure

ratio for the shock wave, defined as P = (P2 −P1)/(P2 +P1); A represents the

Atwood number for the contact, A = (ρ2 − ρ1)/(ρ2 + ρ1). P and A are used

as the wave strength for the shock and the contact.

Shock Contactvariable Left Right Left Right

ρ 3.973980 1.0 1.0 10.0p 1.337250 0.001 0.001 0.001v 1.0 0.0 0.0 0.0γ 1.67 1.67 1.67 1.67

P = 0.999 A = 0.82

Table 2.1: Parameters that define the base case for the shock contact interac-tion.

16

This base case coincides with the base case for the shock contact inter-

action to be studied in Sec. 2.2. We further specify the wall location as 1.5

units to the right of the initial contact location. The transmitted shock, after

interaction with the contact, progresses to interact with (i.e. reflect off) the

wall. This interaction will also be studied in Sec. 2.2. Subsequently, there are

a number of reverberations, of reflected rarefactions and compression waves,

between the contact and the wall and between a new contact formed by a shock

overtake interaction and the original contact. The new contact is clearly visi-

ble in Fig. 2.1 (left), as the vertical line near the left border, starting at a time

about t = 5.2. The interactions are illustrated by the space time contour plots

of the density, shown in Fig. 2.1, left and pressure contours, right. In Fig. 2.2

(left), we show the type and location of the waves, as determined by our wave

filter analysis program. Both figures refer to the base case. The build up of

complex wave patterns is evident. Ten Riemann problems are extracted from

the complex wave interaction problem. A schematic representation of these

ten Riemann problems is given in Fig. 2.2 (right).

The ensemble of 200 initial conditions is defined by a Latin hypercube

variation shock and contact strength by ±10% about the base case defined as

above.

2.2 The Statistical Numerical Riemann Problem

The SNRP introduces errors (modelled as random) in addition to prop-

agating errors or uncertainty from input to output. The waves in the SNRP

have a finite width and the solution algorithm in the SNRP has only finite

17

Figure 2.1: Left. Space time density contour plots for the multiple waveinteraction problem studied in this section. Right. Pressure contour plots forthe base case considered here.

accuracy. Because of the possible finite width of the input waves, the problem

and its solution are not strictly scale invariant, and so we consider a general-

ization of the Riemann problem.

A statistical distribution of numerical incoming waves and starting states

determines the SNRP. Its solution gives the output waves, each of which gen-

erates the same type of data Thus we define the SNRP as a statistical (non-

deterministic) mapping from a statistical input wave description to a statistical

output wave description.

The statistics of the SNRP mapping function arise from grid errors, and

from the random placement of a travelling wave relative to the centers of the

finite difference lattice. Our objective in this section is to build up a library

18

Figure 2.2: Left: Type and location of waves as determined by our wave filteranalysis for the base case considered here. Right: Schematic representationof the waves and the interactions, with labels for the interactions, taken fromthe left frame.

of statistical input-output relations that will include all Riemann problems to

be encountered in Sec. 2.3. This library will be used to predict results for the

multi-wave error and uncertainty analysis based on a multi-path scattering

formula.

2.2.1 The Isolated Shock and Contact Wave

We start with the analysis of the ensemble averaged mean width of a

single (non-interacting) wave.

Fig. 2.3 (left) shows the expected narrow and time independent (∼ 2∆x)

shock width. Among the several factors contributing to wave strength and

19

speed errors, we mention the finite accuracy of the Riemann solution root

solver used in the numerical scheme, and the numerical (finite difference) na-

ture of the solution. The latter arises in two ways, the relaxation to a constant

ambient state and the finite rate of convergence under mesh refinement, both

applicable on the post shock or up stream side of the shock wave.

Figure 2.3: Ensemble mean shock width and the standard deviation of theshock width (left frame). The mean width, equal to about 2∆x, is much largerthan the standard deviation, indicating that the mean width is essentially adeterministic feature of the solution. Convergence properties of the travellingwave to the steady state values on each side of the wave (right frame). Thestraight line in the right frame is the asymptote to the exponential convergencerate, with slope 0.01 in units of ∆x.

According to the theory of travelling waves for the viscous Riemann prob-

lem [40], considered as a model for numerically generated travelling waves, we

expect an exponential approach of the numerical shock profile to its limiting

values at x = ±∞. The error occurs on the upwind side of the shock, while

20

the down wind states converge identically to their far field values within a few

mesh blocks. See Fig. 2.3 (right). We measure the local extrema in the error

dimensionlessly as emax = |(ρ − ρ∞)/ρ∞| where ρ∞ is the far field density.

Then we model emax(n∆x) = c exp(−λn) where n is the distance from the

shock front in mesh units. We find λ = 1.0 × 10−2 and c = 2.4 × 10−4 for

the shock wave defined in the base case, see Table 2.1. The first extrema is a

local maximum, occurring about 4∆x from the center of the shock front. The

details of the shock error behavior will be sensitive to the numerical method,

but the general form of the error, as it is derived from a mathematical theory,

should be somewhat universal.

Fig. 2.4 (left) shows the larger contact width wc ∼ cct1/3 growing from 2

to 30 cells with a rate asymptotically proportional to t1/3. Similar asymptotics

have been observed by Harten [25] for an ENO scheme. The rate t1/3 results

from the second order accuracy of the method used here. The numerical

diffusion is sensitive to the direction of mixing. For the flow from high density

to low, the numerical mixing is of heavy fluid into light. We call this the step

down problem. For the step down problem, shown in Fig. 2.4 (left), we find

cc ∼ 1,

The reverse, called the step up problem, flows from light to heavy fluid. It

mixes small amounts of light fluid into heavy, an effect less noticeable in terms

of the diffusion width, especially for large density contrasts. For a step up

flow, we find wc ∼ min5, cct1/3. See Fig. 2.4. As a partial explanation of this

difference between the step down and the step up problems, we note that the

spreading is primarily associated with the up stream side of the contact, and

21

Figure 2.4: Ensemble mean contact width for isolated noninteracting waves.Because the width is entirely grid related, we record width in units of ∆x andtime in units of the number of time steps. The standard deviations are alsoplotted, and are the points to the extreme left in each frame. Left (step down):we observe an increase from 2 cells to 30 over 104 steps and an asymptoticgrowth rate cct

1/3, where cc ∼ 1 depends on the flow Mach number. Thestraight line in the left frame is the asymptote to the contact width, withslope 3. Right (step up): We observe a bound on the contact width.

that continued spreading (the t1/3 asymptotics) depends on the up stream flow

being subsonic. The higher sound speed in the light fluid gives a supersonic

upstream state for the step up problem but not for the step down problem, for

the flow parameters considered here. These properties appear to be sensitive

to the details of the numerical algorithm, and specifically to the form of the

limiter employed. We have used a MUSCL algorithm here.

For some aspects of the solution error, the probabilistic error formalism

is more general than is required. When the standard deviation of the error

is much smaller than the mean error (when the coefficient of variation, their

22

ratio, is close to zero), then the error is essentially deterministic, and the

probabilistic formulation is unnecessary. This is the case in Fig. 2.4, with the

standard deviation of the width, shown to the left scale of Fig. 2.4, significantly

smaller than the mean width.

2.2.2 Shock Contact Interactions

We study wave strength, width and position errors after the numerical

shock-contact interaction, the wave interaction of a shock wave moving (to the

right) into a contact (density increase, or step up case). This wave interaction

initiates the complex series of interactions to be studied in Sec. 2.3. The base

case for this interaction is also defined in Sec. 2.1, see Table 2.1. The shock

strength is given with a pressure ratio 1337 (P = 0.999), and the contact

strenth is given with a density ratio 10 (A = 0.82). The equation of state

is a gamma law gas, with γ = 1.67. In Fig. 2.5 (left), we show the time

development of the widths of the two incoming waves and the three outgoing

waves for the shock-contact (step up) interaction. This interaction occurs early

in the development of the contact. In Fig. 2.5 (right), we show the position

errors as a function of time. We adjust the ensemble to have a common time

and location of interaction.

We represent the wave properties as a quadruple

wak = (ωa

k , λak, s

ak, p

ak) , (2.1)

where ω is a wave strength, λ is a wave width, s is a wave speed error, and

23

Figure 2.5: Left: ensemble mean shock and contact widths before and after ashock contact (step up) interaction and standard deviations. Right: ensemblemean shock and contact position errors as a function of time, expressed in gridunits. Step up case.

p is a position error. Also a = i for input wave and a = o for output wave.

We choose dimensionless variables to measure the wave strengths: a modified

Atwood number A = (ρ2 − ρ1)/(ρ02 + ρ0

1) to measure the contact strengths,

and a similar expression built out of the pressures, P = (P2 − P1)/(P02 + P 0

1 ),

for the other wave types. Here the quantities ρ0i and P 0

i denote densities

and pressures from the base case associated with the ensemble (we specify

a starting state for the incoming waves and a base case for the strong wave

strengths). We consider variation about this base case by ±10% in the density

ratio (for contacts) or pressure ratio (for all other waves). The wave widths

are measured in units of mesh spacing. The wave position errors are specified

24

in mesh units, after an isolated transient period. Within this formulation, we

can describe the output wave errors by an expression linear in the two input

wave strengths, i.e. linear in the product of the input wave strengths.

For input wave strengths wi1, wi

2 (i ≡ in) and output wave strengths wo1,

wo2 and wo

3 (o ≡ out) (ordered from left to right), the multinomial expansion for

the output is defined by its coefficients αk,J , for J a multi index, J = (j1, j2).

The expansion has the form

wok =

∑J

αk,Jwi,J , (2.2)

where wi,J = (wi1)

j1(wi2)

j2 . The coefficients αk,J depend parametrically on

the base case Riemann problem, about which a specified variation is allowed.

Given a statistical ensemble of input and output values wi and wo, we use a

least squares algorithm to determine the best fitting model parameters αk,J ,

for any given polynomial order of model. We use (2.2) variationally, that

is to map input variation (about the base case for the ensemble) to output

variability. In other words, (2.2), which is a formula for wave strengths, implies

a similar formula with different but computable coefficients αk,J , in which

all ω’s are defined as variations from the base case, so that they represent

uncertainty or error. We consider the case of a linear input-output relation,

ωok = αk,0 +

∑j αk,jω

i,j. We have an expansion similar to (2.2) for wave widths

and wave position errors.

We begin with the analysis of the SNRP at the ensemble averaged level.

We present the mean model analysis in Table 2.2, with ±10% variation about

25

the base case. The input contact width has been set to zero, as part of the

specification of this SNRP.

variable \ coef const ωi1 ωi

2 error(r. sonic) (contact) L∞ STD

ωo1 (l. sonic) -0.208 0.454 0.251 0.47% 0.001

ωo2 (contact) -0.042 0.000 0.912 0.03% 0.0001

ωo3 (r. sonic) -0.286 1.004 0.346 0.30% 0.001

λo1 (l. sonic) 2.184 -0.563 0.000 122% 0.240

λo2 (contact) 4.725 0.110 -1.466 0.67% 0.010

λo3 (r. sonic) 2.197 0.068 0.106 5.35% 0.057

po1 (l. sonic) 0.221 -0.014 0.023 27.1% 0.022

po2 (contact) 0.426 0.001 -0.092 1.78% 0.002

po3 (r. sonic) 0.332 -0.004 -0.099 3.47% 0.005

Table 2.2: The SNRP shock contact (step up) interaction. Expansion coeffi-cients for output wave strengths, wave widths and wave position errors (linearmodel) for input variation ±10%. Here the base case input contact wave widthis zero.

To read Table 2.2, we note that the first (wo1) row (labelled in the table

as wo1 (l.sonic)) lists coefficients α1,J for J = (0, 0), J = (1, 0), etc. These

coefficients are determined by a least squares algorithm, that minimizes the

expected, or mean error over the ensemble, in comparing the linear predictions

to the exact solution of the Riemann problem. The mean is taken over the

statistical ensemble. The last two columns describe errors in the model (2.2).

The column labelled L∞ is the maximum of the absolute value, over the en-

semble, of the relative error, expressed in per cent. It is thus a pointwise error

estimate. The maximum is computed using a sample size of 200, and it may

be sensitive to the choice of ensemble. The relative error is defined as (pre-

dicted - exact)/exact where exact is the result of the (exact) Riemann solution

26

and predicted is the value given by the finite polynomial (linear) model. The

column STD is the standard deviation of (predicted - exact). From the small

values of these errors for the linear model, as seen in Table 2.2, we conclude

that the linear model is adequate for many purposes.

The three variable (λ) rows in Table 2.2 represent wave width errors.

The large sup norm error for the width of the reflected shock results from

a few outliers, mostly but not entirely due to smaller shock widths than the

mean. The standard deviation for this quantity is about 10% of the mean

value, indicating that the error model is (on the whole) satisfactory, and that

the shock wave widths are not (mostly) fluctuating greatly. The outliers are

mainly associated with time steps and realizations for which the (narrow)

reflected shock has at most one internal mesh points. For these cases, our

filter tool for assessing the numerical shock width and position is not effective,

so the outliers can be viewed as a breakdown of the diagnosis methodology.

We also study the wave position errors. Fig. 2.5 (right) shows the position

errors as a function of time. Following the initial transient, the position error

is constant, reflecting convergence of the wave speed to its exact value. The

errors in the wave position rows of Table 2.2 present this constant error, given

in mesh units. All position errors are subgrid. The standard deviations are

smaller than the means, indicating that the errors are basically deterministic.

The L∞ position error is the supremum of the relative error. It is an error

in the model of the error, i.e. the error in the error. Occasional ensemble

members with very small (exact) position error produce a small denominator

in the relative error, (model error)/(exact error). Thus the large entries in

27

this column (also in other tables) do not represent a deficiency in the error

model. We see similar and more extreme L∞ per cent errors in later tables,

with the same cause. Note that the standard deviation is comparable to the

mean position error, so that occasional instances of nearly zero error are to be

expected.

2.2.3 Shock Crossing Shock Interactions

Here we study the reflection of the shock off the wall, a special case of

the shock crossing shock interaction. We can ignore the shock wave width

parameters, as these are narrow and deterministic. We only consider one

output wave position error, po1. See Table 2.3. The shock wave strength errors

(the far field, large separation errors) are small.

variable \ coef const ωi1 error

(r. sonic) L∞ STDωo

1 (l. sonic) -0.002 0.716 0.014% 0.000032λo

1 (l. sonic) 2.291 -0.422 7.923% 0.062po

1 (l. sonic) 0.060 -0.039 5065% 0.009ωo

2 (contact) 0.057 0.0003 24.4% 0.005λo

2 (contact) 5.9 50% 0.7

Table 2.3: The SNRP defined by the crossing of two shocks. Expansion coef-ficients for output wave strengths, widths and position errors (linear model)for input variation ±10%.

The contact mode contributes an error, well known as shock wall heating.

The exact solution has no contribution in this mode for a wall reflection.

The error is the imprint on the entropy, temperature and density variables of

28

entropy errors made during the shock interaction process, apparently due to

shock oscillations. Since entropy can only increase, according to the second law

of thermodynamics, these errors do not cancel. Because the solution algorithm

conserves mass locally, we expect the spatial integral of the density errors

to cancel approximately. Since the velocity of the fluid at the wall is zero,

these errors do not move, and remain permanently attached to the wall. As

a numerical wave, the error is a standing wave. We do not have a theoretical

model for the form of these errors. Due to this lack, our fitting of the errors

will be less precise than those discussed elsewhere in this paper. We define the

wall error width to be the distance to the wall in mesh units of the furthest

location for which the density error is at least twice the background noise in

the post shock region, or about 1% of the base case density. This width is

about 6∆x. Also the wave strength error is defined dimensionlessly as the L1

error in density, divided by ρ0∆x, where ρ0 is the base case post shock density

after the wall reflection.

2.2.4 The Contact Reshock Interactions

After reflection from the wall, the transmitted lead shock wave re-crosses

the deflected contact. This is a step down interaction. We have one input

and one output wave width parameter, both for the contact. According to the

analysis of Sec. 2.2.1, the contact width is is modelled as cct1/3 where both

the width and t are expressed in mesh units. This formula is accurate after

some 50 time steps, according to Fig. 2.5, and the Table 2.4 entry λo2 = cc in

this formula. We form a linear model for this constant in this expression in

29

Table 2.4. The rarefaction width has the form constant + rate × time. We

find very small errors in the rate, not tabulated here. The entry λo3 refers to

the constant, which gives an offset for the centering of the rarefaction wave.

This entry is expressed in mesh units.


2 error(contact) (l. sonic) L∞ STD

ωo1 (l. sonic) 0.282 -0.314 0.645 0.57% 0.0008

ωo2 (contact) 0.013 0.819 0.118 0.20% 0.0003

ωo3 (r. sonic) -0.128 0.143 0.468 0.41% 0.0004

λo1 (l. sonic) 2.383 0.754 -1.307 5.47% 0.038

λo2 (contact) 0.909 0.011 0.216 1.00% 0.005

λo3 (r. sonic) 3.619 0.151 -0.974 14.8% 0.138

po1 (l. sonic) 0.242 0.043 0.042 10.4% 0.014

po2 (contact) -0.036 0.045 0.066 75.5% 0.008

po3 (r. sonic) -0.447 0.078 -0.036 16.7% 0.029

Table 2.4: The SNRP shock contact (step down) interaction. Expansion co-efficients for output wave strengths, widths and position errors (linear model)for input variation ±10%.

2.3 The Composition Law

Here we add up all the pieces. We introduce a formula for combining the

wave interaction errors defined in Sec. 2.2 for isolated Riemann problems, to

yield the error for arbitrary points in the complex wave interaction problem.

The formula is validated for fully resolved simulations and it is shown to be

partially correct and partially incorrect for under resolved simulations.

30

2.3.1 A Multipath Integral for a Nonlinear Multiscat-

tering Problem

We begin with a formula expressing the error in a given Riemann problem

R0 as multinomial expansion associated with initial waves and errors located

inside its domain of dependence. For a 1D shock wave interaction problem,

think of the solution as being primarily composed of localized waves, interact-

ing through Riemann problems and generating outgoing waves, that further

interact in the same manner. The interaction of waves generates a planar (1D

space and time) graph, the vertices of which are the Riemann problems and

the bonds are the travelling waves, between Riemann problem interactions.

Starting from a given Riemann problem (vertex) or wave (bond), we can trace

backward and determine its domain of dependence. Call this graph G.

For each Riemann problem, we consider two types of vertices, correspond-

ing to the constant and linear terms in the parameterized approximate solution

and error terms developed in Sec. 2.2. We treat the linear terms separately,

as they allow a simple propagation law,

SL =

∫w(t = 0)dω (2.3)

where w(t = 0) is a vector representing the strength of the time zero wave and

its error or uncertainty, evaluated at the beginning of the path ω, and SL is the

purely linear propagation contribution to a final time error. The path space

integral dω is taken over all paths progressing in time order through G from

the initial time to the final vertex, with each term weighted by the appropriate

31

linear factors from the formula for the approximate solution of the Riemann

problems transversed. This path space representation makes evident the point

that the solution SL is that of a multiple (linear) scattering problem.

Figure 2.6: The solution and its errors at the point (x, t) can be obtainedby “adding up” the solution and errors for the waves within the domain ofdependence

The amplitude S at the final time (vertex of G) can similarly be thought

of as a solution of a nonlinear multiple scattering problem, leading to a repre-

sentation in terms of multipath integrals. To allow nonlinear (constant) inter-

actions, we re-introduce the vertices from these other terms. Let V = V(G) be

the set of vertices of G, and let B ⊂ V be a subset of V where constant terms

occur. The total amplitude S will then be a sum over terms SB indexed by B.

For each v ∈ B, let Iv be the the interaction coefficient, taken from a table of

Sec. 2.2. We write

S =∑

B⊂V(G)

SB =∑

B⊂V(G)

∫ ∏v∈B

IvdωB . (2.4)

32

The multipath propagator dωB is a product of the individual propagators ω

for each single path, as in (2.3). The summation in (2.4) can be understood

schematically as the sum over all events within the domain of dependence of

the evaluation point (x, t) at the vertex of G. See Fig. 2.6.

2.3.2 Evaluation of the Multipath Integral

From prior work [7], we know that the dominant errors in the composite

solution are located within the leading shock and contact waves of the problem.

A portion of these errors are simple resolution errors (created errors at the

interaction). Wave strength errors can be modelled as only transmitting from

the initial uncertainty. Wave width errors shows up in the created errors only.

As for wave position errors, transmitted errors from the previous interaction

and created errors at current interaction should be combined together as total

position errors.

We develop formulas for the propagation of the initial uncertainty to the

output uncertainty of interaction 3, in Fig. 2.2 (right). The initial uncertainty

is reflected in the wave strength variables, according to the definition of the

ensemble of initial conditions. The transmission of the mean values through a

linear model is standard and is not detailed here. The transformation of the

variance is our major concern. Let B(l) with matrix entries β(l)jk be the matrix

that gives the linear transformation of these variables due to interaction l. We

note that the matrix entries β(l)jk are defined by the ωi columns and ωo rows of

Table 2.2 (linear; l = 1), Table 2.3 (l = 2), and Table 2.4 (l = 3). The output

to interaction 3 has three components, and we compute the variance of each,

33

labelled j = 1, 2, 3. We have

Var ωo(3)j = (β

(3)j2 )2Var ω

i(3)2 = (β

(3)j2 )2Var ω

o(2)1

= (β(3)j2 )2(β

(2)11 )2Var ω

i(2)1 = (β

(3)j2 )2(β

(2)11 )2Var ω

o(1)3

= (β(3)j2 )2(β

(2)11 )2

2∑k=1

Var (β(1)3k )2ω

i(1)k (2.5)

We also need to calculate formulas giving the transmission of position

errors through the various Riemann problems. For interaction with a wall (e.g.

case 2), the formula is elementary. Assuming no error in the wall position, let

pi and po denote input and output position errors for a reflection off of a

stationary wall, where the output is due transmission of error, i.e. due only to

the input, as opposed to Sec. 2.2, where there is no input position error and

the output position error is created during the interaction. Then we have

po = pi vo

vi(2.6)

where vi and vo are the incoming and outgoing wave speeds for the waves

involved in the wall reflection. For the interaction of two incoming waves, the

result is slightly more complicated. Each of the two terms in the formulas

below is due to the input error in one of the input waves. That error can be

computed by (2.6) if we perform the analysis in the frame in which the other

wave is stationary. The result is

po1 =

pi1(v

o1 − vi

2) + pi2(v

i1 − vo

1)

vi1 − vi

2

(2.7)

34

po2 =

pi1(v

o2 − vi

2) + pi2(v

i1 − vo

2)

vi1 − vi

2

(2.8)

po3 =

pi1(v

o3 − vi

2) + pi2(v

i1 − vo

3)

vi1 − vi

2

(2.9)

where pij is the position error of the incoming contact wave (j = 1) or left facing

shock (j = 2). The complete position error model is obtained by adding the

results of (2.7) - (2.9) to those of Table 2.4 for the position errors created at

the interactions. For interactions 1 - 3, we model the wave width (error) as a

created error only.

2.4 Numerical Results

2.4.1 Errors in Fully Resolved Calculations

We regard a calculation as resolved if all (the principal) waves have sep-

arated, with converged left and right asymptotic states, before they interact

with another wave. For this type of simulation, we choose 500 mesh cells in

our basic simulation study. We examine errors in wave strength, wave position

and wave width, based on the graphical expansion given in Sec. 2.3.1, 2.3.2.

The wave strength errors are dominated by the transmission of error (or un-

certainty) from the initial conditions. In Tables 2.5, 2.6 and 2.7 we compare

the predicted error with the error computed directly, taken from a full solution

of the multiple wave interaction problem. The model for the prediction of the

error is satisfactory for all cases: the wave strength and its errors, the wave

width errors, and the wave position errors.

35

variable \ error Simulation Predictionmean wave strengths

ωo1 (l. sonic) 0.451 0.452

ωo2 (contact) 0.704 0.703

ωo3 (r. sonic) 0.999 0.998

wave strength errorsVar ωo

1 (l. sonic) 0.0008 0.0008Var ωo

2 (contact) 0.0019 0.0018Var ωo

3 (r. sonic) 0.0035 0.0036wave width errors

λo1 (l. sonic) 1.630 1.622

λo2 (contact) 3.636 3.635

λo3 (r. sonic) 2.346 2.352

wave position errorspo

1 (l. sonic) 0.220 0.226po

2 (contact) 0.313 0.312po

3 (r. sonic) 0.200 0.202

Table 2.5: Predicted and simulated errors for output wave strengths, wavewidths and wave positions, Interaction 1.


ωo1 (l. sonic) 0.713 0.714


1 (l. sonic) 0.0018 0.0018wave width errors

λo1 (l. sonic) 1.868 1.869


1 (l. sonic) -0.118 -0.092


36


ωo1 (l. sonic) 0.520 0.519

ωo2 (contact) 0.674 0.674

ωo3 (r. sonic) 0.306 0.305


1 (l. sonic) 0.0009 0.0010Var ωo

2 (contact) 0.0012 0.0013Var ωo


λo1 (l. sonic) 2.097 1.982

λo2 (contact) 5.027 4.918

λo3 (r. sonic) 2.875 3.033


1 (l. sonic) -0.097 -0.105po

2 (contact) -0.003 0.013po

3 (r. sonic) -0.151 -0.134


37

2.4.2 Errors in Under Resolved Calculations

Here we allow 100 cells for the coarse grid simulation. This resolution

allows 10 cells between the contact and the reflection wall at the time of

interaction 3 and beyond. Since the contact has a width of 5 cells, since the

right facing rarefaction is about this size and since the wall has inaccurate

states in a region of several mesh blocks neighboring it, we are clearly at the

limit of the present diagnostic methods based upon the wave filter. For the

same reasons, the calculation is clearly under resolved. For this reason we are

not able to analyze data for the case of a coarse grid simulation with 50 cells

using the present version of our wave filter. Again we present the first three

interactions in detail, at 100 cell resolution, comparing the predicted to the

directly simulated errors. See Tables 2.8, 2.9, 2.10. We see good results for the

wave strengths and their errors and for the wave width errors, and poor results

for the comparison of position errors. This can be understood in terms of the

decay time for convergence to asymptotic large time values for the position

errors, an explanation that also accounts for the difference with the resolved

case, for which the simulated and predicted position errors agree. The position

errors have a relatively slower decay time. The other three quantities show a

high level of agreement between the resolved and under resolved cases. The

wave width error is expressed in grid units, and so should be the same in the

two cases differing in grid resolution only. For the wave strength entries, the

lack of dependence on grid resolution is due to the fact that these quantities are

dominated by the uncertainty expressed in the ensemble of initial conditions,

which is independent of grid resolution.

38


ωo1 (l. sonic) 0.451 0.452

ωo2 (contact) 0.741 0.703

ωo3 (r. sonic) 0.996 0.998


1 (l. sonic) 0.0008 0.0008Var ωo

2 (contact) 0.0022 0.0018Var ωo


λo1 (l. sonic) 1.381 1.621

λo2 (contact) 3.498 3.635

λo3 (r. sonic) 2.347 2.352


1 (l. sonic) 0.972 0.226po

2 (contact) 1.539 0.312po

3 (r. sonic) 0.785 0.202

Table 2.8: Case 1. The contact-shock interaction (step up). Errors for outputwave strengths, wave widths and wave position. Comparison of under resolvedsimulation and prediction.


ωo1 (l. sonic) 0.721 0.712



λo1 (l. sonic) 1.718 1.871


1 (l. sonic) -0.401 -0.092

Table 2.9: Case 2. The shock crossing equal shock (wave reflection) interaction.Errors for output wave strengths, wave width and wave position. Comparisonof under resolved simulation and prediction.

39


ωo1 (l. sonic) 0.523 0.514

ωo2 (contact) 0.669 0.705

ωo3 (r. sonic) 0.318 0.315


1 (l. sonic) 0.0009 0.0010Var ωo

2 (contact) 0.0013 0.0013Var ωo


λo1 (l. sonic) 1.964 2.000

λo2 (contact) 4.606 4.928

λo3 (r. sonic) 2.169 3.029


1 (l. sonic) -0.551 -0.103po

2 (contact) 0.301 0.015po

3 (r. sonic) 0.432 -0.131

Table 2.10: Case 3. The contact-shock interaction (step down). Errors foroutput wave strengths, wave width and wave position. Comparison of underresolved simulation and prediction.

40

Chapter 3

Error Analysis for Spherical Geometry

3.1 Introduction

We are concerned with the identification and characterization of solu-

tion errors in spherically symmetric shock interaction problems. This issue

applies to the study of supernova and the design of inertial confinement fusion

(ICF) capsules. In the first case, theory and simulations contain a number of

uncertainties, and comparison to observations is thus not definitive. A sys-

tematic effort to remove some of the uncertainties associated with simulation

will thus be a useful contribution. In the second case of ICF design, concern

over solution accuracy has led to mandates of formal efforts to assure solution

accuracy.

In Chapter 2, we have analyzed shock interactions in a planar geometry

[7, 11], following the general approach to uncertainty and numerical solution

error developed in [22, 23]. Here we focus specifically on complications which

result from spherical geometry. In brief, these are:

1. The solution waves are not of constant strength between wave interac-

41

tions, but evolve approximately according to a power law as a function

of the radius.

2. The solution is not spatially constant between waves.

3. If the solution is not required to be spherically symmetric, the prob-

lem of identifying wave structures as curves or surfaces in 2D or 3D is

introduced.

These are classical problems as far as the solutions are concerned, but

the application of these ideas to the analysis of errors in the solution appears

to be new. The radially dependent strength of spherical waves is discussed

in [40]. The spatial variation of spherical waves is contained in the Guderley

solution [24].

The problem of analysis of errors in numerical solutions is of course central

to numerical analysis. Much of this effort is motivated by other concerns, and

appears not to be directly applicable to the problems we address.

As a technical introduction to this chapter, we study errors of a spherical

shock interaction problem (shown in Fig. 3.1) on uniform radial grid of 100

and 500 cells (with errors determined by reference to a 2000 cell calculation

referred to as the fine grid). The base case for each wave interaction coincides

with the base case assumed for the interactions studied in Sec. 3.2. The trans-

mitted shock, after interaction with the contact, progresses to interact with

(i.e. reflect off) the origin. Subsequently, there are a number of reverberations,

of reflected rarefactions and compression waves, between the contact and the

origin. We use MUSCL [5] as the numerical method; for the comparison of

42

Figure 3.1: Left: space time density contour plot for the multiple wave in-teraction problem studied in this chapter, in spherical geometry. Right: typeand location of waves determined by the wave filter analysis with labels forthe interactions. Here I.R. denotes the inward moving rarefaction.

tracked to untracked solutions of the problem, see [8]. The equation of state is

a γ-law gas with γ = 5/3. The ensemble of 200 initial conditions is defined by

a Latin hypercube variation shock and contact strength by ±10% about a base

case defined (as in Sec. 2.1) by a contact located at 1.5 units from the origin;

an inward moving shock located 2.25 units from the origin, with all constant

states between waves. The initial base case shock strength is M = 32.7 and

the initial base case Atwood number for contact is 0.82.

3.2 The Statistical Numerical Riemann Problem

In this section, we study statistical numerical Riemann problems (SNRP)

in spherical geometry.

43

3.2.1 The Single Propagating Wave

We start with the analysis of the single propagating inward shock in

spherical geometry. The radially dependent strength of convergent spherical

shocks is discussed in [40]. The spatial variation of convergent spherical shocks

is contained in the Guderley solution [24]. The inward moving shocks are not of

constant strength as in a planar geometry, but evolve approximately according

to a power law as a function of the radius. From Whitham’s approximation

approach, we have

M ∝ r−1/n, (3.1)

for cylindrical shocks, and

M ∝ r−2/n, (3.2)

for spherical shocks. Here M is the Mach number of the shock, n = 1 + 2γ

+√2γ

γ−1, and γ is the adiabatic exponent defined as the ratio of two specific heats.

A comparison with the exponents from Guderley’s exact similarity solution is

given in Table 3.1.

Cylindrical Sphericalγ Approximate Exact Approximate Exact

6/5 0.163112 0.161220 0.326223 0.3207527/5 0.197070 0.197294 0.394142 0.3943645/3 0.225425 0.226054 0.450850 0.452692

Table 3.1: Comparison of the exponents from the approximate and the exactsimilarity solutions for an inward propagating spherical shock wave.

We also have a similar approximate power law for the shock velocity

44

Figure 3.2: Left. Mach number vs. radius for a single inward propagatingshock. Right. The same data plotted on a log-log scale.

[12, 34]. Fig. 3.2 shows the exponential divergence of the shock strength (here

characterized by the Mach number) at r → 0. The accuracy is amazing

in view of the simplicity of the approximate theory. The figure shows that

converging shocks are reacting primarily with the geometry, as assumed in the

approximate theory, and are affected very little by further disturbances from

the source of the motion; the strength of the initial shock enters only through

the constants of proportionality in (3.1) and (3.2). This is not true for outward

moving shocks. They slow down due to both the expanding geometry and to

the continuing interaction with the flow behind. From Fig. 3.3, however, we

find that the strength of an outward moving shock also follows a power law

which is similar to (3.2) but with a modified exponent, after the radius of the

outward moving shock is three times the initial radius. To develop a model for

shock wave propagation which has a smaller pre-asymptotic regime, we allow

45

two distinct exponents,

M ∝

ra1

ra2

r0 ≤ r ≤ 3r0

r ≥ 3r0.(3.3)

Here we choose a1 = −0.4, a2 = −1.0 for γ = 1.67, and r0 is the initial shock

radius.

Figure 3.3: Left. Mach number vs. radius for an outward moving shock wavestarting at different radii r0. Right. The same data plotted on a log-log scale;the dashed lines in this plot represent the power law model (3.3).

We also study the single propagating contact (step up and step down

cases). Fig. 3.4 shows the contact width wc ∼ cct1/5 growing from 2 to 5 cells

with a rate asymptotically proportional to t1/5. We found that the step up

contact and the step down contact have the same behavior.

46

Figure 3.4: Ensemble mean contact width for a single propagating contact.We record the width in units of ∆x. The standard deviation is also plotted,as the points to the extreme left.

3.2.2 The Shock Contact Interaction

We study the wave strength, speed, width and position errors after a

wave interaction.

We begin with the analysis of the initial shock contact SNRP at the

ensemble averaged level. We present the linear model coefficients in Table 3.2,

with ±10% variation for the initial contact strength and ±5% variation for

the initial shock strength (consistent with ±10% variation in pressure ratio

as used in Chapter 2, [11]) about the base case. According to the analysis

of Sec. 3.2.1, the strength of this initial inward shock is not constant, and is

increasing as it moves toward the origin. We use the power law M = Cr−2/n to

47

estimate the initial shock strength at the interaction time and use this quantity

represented by the variable C as the input shock strength in the modeling. The

input contact width has been set to zero, as part of the specification of this

SNRP.


2 model error(contact) (l. sonic) STD STD/ωo

wave strengths (100 cells)ωo

1 (l. sonic) -33.353 19.521 2.501 0.860 0.954%ωo

2 (contact) 0.374 0.200 0.0003 0.042 7.650%ωo

3 (r. sonic) 3.568 0.402 -0.045 0.009 0.463%wave strength errors (100 cells)

δo1 (l. sonic) -2.039 -3.200 -0.01 0.157 0.174%

δo2 (contact) 0.236 0.016 -0.002 0.021 3.825%

δo3 (r. sonic) 0.053 0.003 -0.001 0.0008 0.041%

wave width errors (100 cells)λo

1 (l. sonic) 1.675 0.305 0.017 0.085λo

2 (contact) 7.093 0.482 -0.146 0.239λo

3 (r. sonic) 2.829 0.302 -0.024 0.107wave position errors (100 cells)

po1 (l. sonic) -0.247 0.242 0.005 0.009

po2 (contact) 0.643 0.065 -0.011 0.192

po3 (r. sonic) -0.042 0.062 0.004 0.009

Table 3.2: The SNRP shock contact interaction. Expansion coefficients foroutput wave strengths, wave strength errors, wave width errors and wave po-sition errors (linear model) for the initial shock contact interaction. Here thebase case input contact wave width is zero. The final columns refer to dif-ference between the linear model (2.2) and the exact quantity. The errors inrows 4-12 refer to the difference between the numerical solution on 100 cellsand the exact solution using 2000 cells.

To read Table 3.2, we note that the first (wo1) row (labeled in the table

as wo1 (l. sonic)) lists coefficients α1,J for J = (0, 0), J = (1, 0), etc. These

coefficients are determined by a least squares algorithm that minimizes the

48

expected, or mean error over the ensemble, in comparing the linear predic-

tions to the exact solution of the Riemann problem. The last two columns

describe errors in the model (2.2). The presence of outliers was monitored and

the ensemble L∞ norm determined (results not tabulated); occasional outliers

indicate non Gaussian statistics. The model error is defined as (predicted - ex-

act) where exact is the result of the simulation and predicted is the value given

by the finite polynomial (linear) model (2.2). The column STD is the standard

deviation of (predicted - exact). Note that the STD errors, as defined, are di-

mensionful. To aid in interpreting the error magnitudes, we present in a final

column (labeled STD/wo) the standard deviation of the error in the model

divided by the mean value of the variable predicted. This column represents

a fractional (dimensionless) error in the model.

According to the analysis of Sec. 3.2.1, the strength of the output inward

moving shock is modeled as Cr−2/n. This formula is accurate after some time,

and the Table 3.2 entry is wo1 = C in this formula. We form a linear model for

this constant in this expression in Table 3.2. We find very small errors in the

exponent, not tabulated here. We developed a model (3.3) for the strength of

the output outward moving shock in Sec. 3.2.1. Here in our study, we are only

concerned with the first formula in (3.3). The entry wo3 in the table represents

the coefficient multiplying the power term.

The three variable (λ) rows in Table 3.2 represent wave width errors.

The standard deviation for this quantity is about 10% of the mean value,

indicating that the error model is (on the whole) satisfactory, and that the

shock wave widths are not (mostly) fluctuating greatly. The inward moving

49

Figure 3.5: Left: ensemble mean inward/outward moving shock and contactwidths after a shock contact interaction. Right: ensemble mean shock andcontact position errors as a function of time, expressed in grid units. Theassociated standard deviations are extremely small, not shown in the plots.In the legend, C. denotes the contact while I.S. and O.S. are the inward andoutward moving shocks.

shock width decreased about 10% relative to the wave width at the interaction

time, while the outward moving shock width increased about 10%. See Fig. 3.5,

left frame. The contact width is modeled as cct1/5 where both the width and t

are expressed in mesh units. The Table 3.2 entry λo2 = cc in this formula. We

form a linear model for this constant in this expression in Table 3.2.

We also study the wave position errors. Fig. 3.5, right frame, shows the

position errors as a function of time. The entries in the wave position rows

of Table 3.2 present those errors, given in mesh units. All position errors are

subgrid. The standard deviations are smaller than the means, indicating that

50

the errors are basically deterministic.

All solution errors are sensitive to the grid spacing, taken to be 100

computational cells in Table 3.2-3.4. This sensitivity is not extreme. For

example, if the 100 cell model is used to analyze the 500 cell data, the model

errors (STD) approximately double, but remain small.

3.2.3 Shock Reflection at the Origin

Here we study the reflection of the shock off the origin. According to the

analysis of Sec. 3.2.1, the input inward moving shock has infinite strength at

the origin. We used the strength at the radius r = 1 as the initial state and the

input shock strength in the modeling process. We study the wave strength,

wave strength errors, wave width and wave position errors. See Table 3.3.

variable \ coef const ωi1 model error

(l. sonic) STD STD/ωo

ωo1 (r. sonic) -242.394 5.606 1.137 0.468%

δo1 (r. sonic) -3.27 0.031 0.112 0.045%

λo1 (r. sonic) 1.221 0.018 0.099

po1 (r. sonic) 0.474 0.001 0.012

Table 3.3: The SNRP defined by the shock reflection at the origin. Expansioncoefficients for output wave strengths, wave strength errors, wave width errorsand wave position errors (linear model) for input variation ±10%.

We found that the Mach number of the outward moving shock (reflected

shock) was essentially independent of the input variation in Mach number. To

explain this phenomena, we recall that the ambient state ahead of the outward

moving reflected shock is an incoming continuously variable flow. The sound

51

speed ahead of this flow is affine linearly dependent on the strength of the

incoming shock wave, as is the shock speed of the reflected outward moving

shock wave. Thus the outward moving Mach number, as a ratio of two quan-

tities varying affine linearly with the incoming shock strength, has a fractional

linear form in the incoming wave strength. A simple calculation shows that

the variation in the outward moving shock Mach number Mo contains the fac-

tor (1−Mo) and since Mo ≈ 1.2, this small factor suppressed variation in Mo

as a function of Mi, the Mach number of the incoming shock. Thus the Mach

number is not a good measure for the outward moving shock strength. We

choose the pressure behind the reflected shock instead as ωo1 in Table 3.3. The

pressure also follows the power law. The large entries in this row result from

the fact that the (dimensional) pressure (ωo1) is much larger in pressure units

than the Mach number (ωi1).

3.2.4 The Contact Reshock Interaction

After reflection from the origin, the transmitted lead shock wave re-

crosses the deflected contact. The outgoing waves from this interaction consist

of a rarefaction wave propagating toward the origin, a contact and a shock

propagating outward. The region inside of the outward propagating shock, on

both sides of the contact is not piecewise constant, but contains an inward

propagating compression, which eventually breaks to form an inward moving

shock, reaching the origin at interaction 4. This inward moving compression

is generated from the geometrically caused weakening of the outward moving

shock, and is a well recognized aspect of spherical shock wave dynamics. The

52


2 model error(r. sonic) (contact) STD STD/ωo

wave strengths (100 cells)ωo

1 (l. sonic) 0.097 -0.108 0.436 0.031 13.305%ωo

2 (contact) 0.103 -0.192 1.168 0.007 1.116%ωo

3 (r. sonic) 0.988 0.195 -0.225 0.003 0.262%wave strength errors (100 cells)

δo1 (l. sonic) -0.291 0.161 -0.468 0.017 7.296%

δo2 (contact) -0.067 0.142 -0.125 0.006 0.957%

δo3 (r. sonic) -0.030 0.107 -0.0004 0.001 0.087%

wave width errors (100 cells)λo

1 (l. sonic) 9.776 -6.372 5.091 0.484λo

2 (contact) 1.903 0.156 -0.677 0.534λo

3 (r. sonic) 4.088 -1.401 1.549 0.168wave position errors (100 cells)

po1 (l. sonic) 4.782 -3.602 2.372 0.379

po2 (contact) -0.453 0.409 -0.054 0.177

po3 (r. sonic) -0.199 -0.685 3.213 0.052

Table 3.4: The SNRP contact reshock interaction. Expansion coefficientsfor output wave strengths, wave strength errors, wave width errors and waveposition errors (linear model).

shock and the rarefaction interact, and eventually the rarefaction disappears

in this interaction. Here we only follow the waves through the output of in-

teraction 3, and thus avoid much of this interaction. Specifically, we focus on

the inward moving rarefaction and not the inward moving shock. We study

the wave strength, wave strength errors, wave width and wave position errors

resulting from interaction 3. See Table 3.4.

According to the analysis of Sec. 3.2.1, this is a step down interaction

and the contact width is modeled as cct1/5 where both the width and t are

expressed in mesh units. We form a linear model for the coefficient cc in this

53

expression in Table 3.4. The rarefaction width has the form constant + rate

× time. The entry λo1 refers to the constant, which gives an offset for the

centering of the rarefaction wave. This entry is expressed in mesh units.

3.3 Composite Shock Interaction Problems

The main point of this section is to formulate and validate the multipath

scattering formula

S =∑

B⊂V(G)

SB =∑

B⊂V(G)

∫ ∏v∈B

IvdωB . (3.4)

that we developed in Sec. 2.3 ([11]) for analysis of errors. We analyze errors

at the output to interaction 3 directly, comparing the 100 mesh and 500 mesh

simulation to a 2000 mesh, fine grid simulation, here taken as a substitute for

the exact solution. These errors are compared to those generated by adding up

and propagating errors from the input data and from each of the interactions

1 to 3, using the multipath scattering formula. Thus, for example, a position

error as input to interaction 1 is translated geometrically to a position error

for the output to interaction 1 via simple geometric considerations as in [11].

This error is propagated to an input error for interaction 2 through solutions

of radial differential equations. Propagation continues, and yields an error at

the output to interaction 3. See Table 3.5. The wave strength rows present the

result of initial uncertainty propagated to the output of interaction 3 as well

as the accumulation of solution errors. The multipath scattering formula gives

reasonable prediction of error magnitudes in all cases except the wave position

54

errors for the under resolved (100 mesh) simulation. We see that the created

numerical solution errors are important. We also find that a major portion

of the created numerical solution errors come from the second interaction, the

shock reflection interaction. A detailed study of these errors and their relative

importance will be presented in the next section.

variable \ error Simulation Prediction Simulation Prediction100 vs. 2000 mesh 500 vs. 2000 mesh

wave strength errors and propagated initial uncertaintiesδo1 (l. sonic) 0.04±2(0.03) 0.03±2(0.02) 0.01±2(0.02) 0.009±2(0.01)

δo2 (contact) 0.14±2(0.05) 0.12±2(0.02) 0.03±2(0.01) 0.03±2(0.008)

δo3 (r. sonic) -0.02±2(0.02) -0.02±2(0.01) -0.006±2(0.005) -0.007±2(0.004)

mean wave width errors mean wave width errorsλo

1 (l. sonic) 3.04 2.83 2.63 2.72λo

2 (contact) 5.36 6.11 5.56 6.08λo

3 (r. sonic) 2.71 3.04 2.92 2.98mean wave position errors mean wave position errors

po1 (l. sonic) 1.25 0.23 0.12 0.18

po2 (contact) 0.43 0.06 0.05 0.04

po3 (r. sonic) -0.73 -0.15 -0.08 -0.11

Table 3.5: Predicted and simulated errors for output wave strengths, wavewidths and wave positions, output to interaction 3. The inward rarefactionand contact strengths are expressed dimensionlessly as Atwood numbers. Theoutward shock strengths are in the units of Mach number. The width andposition errors are in mesh units. The wave strength errors are expressed asmean ± 2σ where σ is the ensemble STD of the error/uncertainty.

3.4 Error Decomposition

In this section, we will study the relative magnitude of the input uncer-

tainty vs. the errors created within the numerical solution. In more detail,

we wish to understand the contribution of each wave interaction to the errors

55

observed at the end of the simulation. Here, the multipath integral formula is

used assuming independence of errors from different sources.

In Fig. 3.6 we illustrate the distinct terms contributing to multipath scat-

tering formula for analysis of the errors. Each number labelled line segment

is a single term (error propagating path), for the error associated with the

output to interaction 3, in which the shock wave reflected from the origin re-

crosses (re-shocks) the contact. The first two paths indicate the uncertainty

originating with the initial conditions, i.e. with the choice of the ensemble.

This uncertainty propagates through two distinct paths, illustrated by the first

two black lines of Fig. 3.6, to reach the interaction site 3. The first one follows

the shocks, the transmitted shock from the interaction 1 to the origin reflected

shock and back to the contact. The second one follows the the contact from

the lead interaction 1, along the contact until it is reshocked at interaction

3. Next we find two gray lines that represent the errors originating during

the interaction 1, and propagating to the output of 3 through the same two

routes. Finally, we find two paths giving the errors that arise at the shock

origin reflection (interaction 2) and propagate to 3 and in the final path, those

arising during interaction 3 directly.

In Figs. 3.7, 3.8, we present three pie charts representing fractional con-

tribution from each of the six interactions to the error variance for the inward

rarefaction, contact and outward shock, respectively, as output to interaction

3. Note that the two associated with input uncertainty are hatched and the

others are solid gray scales. From these charts, we can infer the relative im-

portance between the input uncertainty and the solution error and determine

56

the contribution of each interaction to the total error variance. We see that for

a 500 cell grid, the dominant error comes from the initial uncertainty, while

for the 100 grid over 75% of the error arises within the numerical simulation.

We also show the contributions of each interaction to the mean value of

the final total error. See Table. 3.6. We only show the values correspond-

ing to Diagrams 3 to 6, as the contribution of the first two diagrams (input

uncertainties) is observed to be zero.

Wave Diagram I.R. C. O.S.Number 100 500 100 500 100 500

3 0.10 -0.01 -0.01 0.001 0.09 -0.0094 0.05 0.009 0.1 0.02 -0.02 -0.0045 -0.05 -0.005 0.006 0.0006 -0.04 -0.0046 -0.07 0.015 0.03 0.01 -0.05 0.01

Total Prediction 0.03 0.009 0.12 0.03 -0.02 -0.007Total Simulation 0.04 0.01 0.14 0.03 -0.02 -0.006

Table 3.6: The contribution of each interaction to the mean value of the totalerror in each of three output waves at the output to interaction 3, for 100 and500 mesh units. Units are dimensionless and represent the error expressed asa fraction of the total wave strength. The last two rows compare the totalof the mean error as given by the model to the directly observed mean error.The columns I.R., C., and O. S. are labeled as in Fig. 3.1, Right frame.

57

Figure 3.6: Schematic graph showing all six wave interaction contributionsto the errors or uncertainty in the output from a single Riemann solution,namely the reshock interaction (numbered 3 in the right frame of Fig. 3.1)of the reflected shock from the origin as it crosses the contact. The numberslabeling the circles refer to the Riemann interactions contributing to the error.The numbers labeling the line segments refer to the different error propagatingpaths.

58

(a) Inward Rarefaction

(b) Contact

(c) Outward Shock

Figure 3.7: Pie charts showing the contribution of each wave interaction dia-gram to the error variance of the wave strength at the output of interaction 3,for a solution using 500 mesh units.

59

(a) Inward Rarefaction

(b) Contact

(c) Outward Shock

Figure 3.8: Pie charts showing the contribution of each wave interaction dia-gram to the error variance of the wave strength at the output of interaction 3,for a solution using 100 mesh units.

60

Chapter 4

Conclusions

We have several main conclusions from the error study.

• We see that a very simple model of solution error is sufficient for the

study of (at least the present instance of) a highly nonlinear problem.

The error is linear in the input wave strengths.

• A composition law for combining errors and predicting errors for compos-

ite interactions on the basis of an error model of the simple constituent

interactions has been formulated and validated. For spherically symmet-

ric shock physics problems, the main new difficulties encountered were

the non-constancy of the solution between interaction events and the

non-constancy of waves and errors between interactions. For a planar

geometry, the errors are constant between interactions, while for a spher-

ical geometry, the errors grow (if the wave which carries them is moving

inward) by a power law in the radius. Similarly outward moving waves

and their errors weaken by a power law.

61

• For planar case, we find that although our formalism allows for statistical

errors in the ensemble that in fact, the dominant part of all errors (ex-

cluding position errors) studied were deterministic, in the sense that the

ensemble mean error dominated the ensemble standard deviation. For

spherical case, the composition model applied to construct the variance

of the error in the wave strength, generally understates the STD by a

factor generally between 1.5 and 2, for causes not presently identified.

Using the model, the total error is a sum of six terms, each corresponding

to a pattern of wave interactions and transmissions. Of these diagrams,

two correspond to initial error, following different transmission patterns,

and four correspond to errors created within the solution and trans-

mitted to the output of the reshock interaction, where the errors are

analyzed. We see that for a 500 cell grid, the dominant error comes from

the initial uncertainty, while for the 100 grid over 75% of the error arises

within the numerical simulation. We could conclude that for coarse grid

simulations, there exists increased importance of created errors.

• For planar case, the wave strength uncertainty which is dominated by

input uncertainty (i.e. the definition of the ensemble), is virtually un-

changed between the highly resolved and the under resolved simulations.

While for spherical case, this is not true. We concluded that for under

resolved spherical simulations, there exists increased importance of cre-

ated errors. The wave width errors are both expressed in grid units, and

are comparable between the two levels of resolution. The wave width

errors evidentially have a rapid relaxation to their asymptotic value.

62

• The primary solution errors created by the simulations are the wave

position errors in the under resolved simulations, on the order of a mesh

spacing. These errors are a transient phenomena, but become frozen

into the calculation as new interactions occur before the transient errors

have diminished. The wave position errors have a slow relaxation to

asymptotic values.

• We find that the wave filter performs well as a diagnostic tool, but that its

limitation (in its present version) lies in assuming well separated waves.

Thus we are limited in the degree of under resolution that we can analyze,

in that all waves must be at least partially separated from one another

before entering into a new interaction.

To the extent that a more detailed modelling of these errors is important,

a more accurate model that includes transient effects will be important. Even

with these limitations, the methods and results appear to be promising, and

should be extended to less idealized problems.

These conclusions are established only under several simplifying assump-

tions, namely restriction to one spatial dimension, use of a simplified (gamma

law gas) equation of state, and consideration of only one numerical method.

Further studies are needed to determine the extent that these conclusions have

a general validity.

63

Chapter 5

Future Work

5.1 2D Shock Wave Interactions in Perturbed Spherical

Geometries

For the future work, we address the much more difficult question of the

perturbed interface problem, with the ensuing instability growth and chaotic

flow. Before we can apply the above methods of statistical error analysis, we

require a numerical solution procedure which is O(1) correct. This simply

stated and seemingly elementary requirement has proven to be surprisingly

difficult for the scientific community. Our present results apply to a planar

geometry. After the first correct RM simulation (to achieve agreement with

experimental data) by FronT ier [27, 29], a three way code comparison, with

experimental data from laser acceleration and a theoretical model all achieved

agreement for a single mode 2D RM instability [28]. This result is very encour-

aging, but the mesh resolution used to achieve it was not, in view of the desire

to compute the much more difficult instabilities in 3D and for fully chaotic (as

64

opposed to single mode) flows. For the related RT chaotic instability in 3D,

extensive studies have shown that most simulation codes compute an insta-

bility growth rate which is below the experimental value by about a factor of

2, while the FronT ier values [9, 10, 13] are consistent with the experimental

range of values, but on the high side. For FronT ier, we could introduce sur-

face tension or mass diffusion to lower the instability growth rate. While for

other simulation codes, adding long wave noise in the initial condition would

be one way to raise the instability growth rate.

5.1.1 Single Mode Perturbed Interface

In Fig. 5.1, we show the density plot for a spherical implosion with a single

mode perturbed interface. We will start with the solution convergence of direct

numerical simulations (DNS) of single mode flow through mesh refinement,

regardless of different numerical algorithms. The convergence should be on

the point-wise sense. Meanwhile, we will determine the solution sensitivity to

mesh size, algorithm, mass diffusion and etc.

We define the L1-error of two different mesh simulations. Suppose ρc(x, t)

is the discrete density field from a coarse grid simulation (as for example

computed on a fixed size Eulerian mesh with grid size measured by h) and

ρf (x, t) is a fine gird solution density. we define the time dependent L1-error

on the computational domain Ω as:

‖ρc − ρf‖L1(t) =

∫Ω

|ρc(x, t) − ρf (x, t)|dx. (5.1)

65

Figure 5.1: Density plot for a spherical implosion simulation with a perturbedinterface (single mode). The grid size is 200 × 200.

For the simulations described in this chapter, the computational domain is

given by Ω = x :√

x · x ≤ Rmax. Since we are assuming axisymmetry for

our flow, the above integral can computed by the formula:

‖ρh − ρf‖L1(t) =

∫Ω

|ρh(r, z, t) − ρf (r, z, t)|rdrdz, (5.2)

5.1.2 Chaotic Mixing

Chaotic flows display a wealth of detail which is not reproducible, nei-

ther experimentally nor in simulations. Generally speaking, this detail is not

relevant, and fortunately, only the statistical averages of the detail are of im-

portance. Thus direct numerical simulation (DNS) of mix, as discussed in

the previous section, gives more information than is needed, and information

66

which in detail cannot be reproducible. Since we really want the averages of

the DNS simulations, the natural question is to find averaged equations which

will compute the averaged quantities directly, without use of the difficult in-

termediate DNS step.

Figure 5.2: Density plot for a spherical implosion simulation with a chaoticperturbed interface (multiple modes). The grid size is 200 × 200.

In Fig. 5.2, we show the density plot for a spherical implosion with a

chaotic perturbed interface. We will analyze the averaged quantities. The

averaged DNS solution should converge under mesh refinement and under

ensemble size approaching infinity. Then we will compare the averaged DNS

solution to the solution of averaged equations [31] (see also earlier work [20, 30]

and references cited there).

Averaged equations arise in many areas of science. Generally, when the

original equations are nonlinear, or when the coefficients of a linear term are

67

to be averaged, lengthy discussions of how to formulate the averaged equations

ensue. The issue is that nonlinearities do not commute with averaging, so the

average of a nonlinear function is not equal to the function evaluated at the

average value of its argument. In addition, the phenomena at a physics level

are much richer, as the averages depend on the averaging length scale. We

wish to average over each phase, and end up with multi-phase flow equations.

The nonlinear closure terms will then reflect the forces, etc., exerted between

the two phases.

From the comparison of averaged DNS solution and solution of the aver-

aged equation, we will be able to examine the adequacy of closures and com-

pose distinct forms of averaging. Also, we will analyze the numerical errors in

solution of averaged equations.

68

Chapter 6

Appendix

6.1 Complete List of Ten Riemann Problems (Planar

Geometry)

For each of the 10 Riemann problems of Fig. 2.2, Right, we vary the wave

strength and for contacts only, we vary the wave width. However, when solved

using the (idealized) Riemann solver, the wave widths are all set to zero. Three

variables out of the nine variables which define three reference states (for two

waves) are not varied in this study. We have two criteria in selecting the

reference variables to hold fixed. If one of the states has a reference ambient

velocity, for example a velocity v = 0 for a state near a wall, we want to

preserve this property and freeze this velocity. For the pressure and density

values, we generally freeze those on the smaller side of the waves, as this gives

a more meaningful variation of the state, uniformly specified as ±10% of the

wave strength, as defined in Sec. 2.2.

Here, we present error models for cases 4-10 in this multiple wave inter-

69

action problem, to complete the analysis of cases 1-3 in Chapter.2.

Case 1: Lead shock interacts with contact

The mid state is held fixed, and the two wave strengths are varied.

Figure 6.1: Problem 1: Shock-contact (step up)

Case 2: Transmitted shock reflects off of wall

The right state is held fixed and the left state is varied.

Figure 6.2: Problem 2: Shock-wall interaction

70

Case 3: Shock reflected off wall recrosses the contact

The right state velocity v = 0.0 is fixed and the left state densities and

pressures are held fixed.

Figure 6.3: Problem 3: Contact-shock (step down)

Case 4: Reflected rarefaction from case 3 reflects off of

wall

Here we study the reflection of the rarefaction off the wall, a special case

of the rarefaction crossing rarefaction interaction. The right state velocity

v = 0.0 is fixed and the left state densities and pressures are held fixed. When

modelled as a SRP, the input rarefaction wave width is set to zero. When

modelled as a SNRP, the input rarefaction wave width is an input parameter.

Similar comments apply to the most of the later cases.

We have one input and one output wave width parameter, both for the

rarefaction. We assume that at interaction location the input rarefaction and

the output rarefaction have the same width. The rarefaction width has the

form

constant + initial width + rate × time (6.1)

71

Figure 6.4: Problem 4: Rarefaction-wall

Here, initial width means the input rarefaction width, which would occur at

the interaction time without the influence of the interaction. We find very

small errors in the rate, not tabulated here. The entry λo1 in Table 6.1 refers

to the constant in formula (6.1), which gives an offset for the width of the

rarefaction wave. This entry is expressed in mesh units. We form a linear

model for this constant in Table 6.1.



1 (l. sonic) 0.146 0.657 0.181% 0.0003λo

1 (l. sonic) 3.832 2.091 5.056% 0.096po

1 (l. sonic) 0.117 -0.008 15.56% 0.006

Table 6.1: Case 4. The SNRP defined by the crossing of two rarefactions.Expansion coefficients for output wave strengths (linear model) for input vari-ation ±10%.

Case 5: Rarefaction reflected off of wall crosses contact

After reflection from the wall, the transmitted rarefaction wave re-crosses

the deflected contact. The right state velocity v = 0.0 is fixed and the left

72

state densities and pressures are held fixed.

Figure 6.5: Problem 5: Contact-rarefaction

We have two input wave width parameters, one each for the contact and

the rarefaction, and three output wave width parameters, one each for the left

rarefaction, the contact and the right compression wave. According to the

analysis of single isolated waves, the output contact width is bounded after

some 100 time steps. The Table 6.2 entry λo2 refers to this bounded width. We

assume that at interaction the input rarefaction and the output rarefaction

have the same width. The rarefaction width has the form (6.1). Here, initial

width means the width, which eon would have at the interaction time without

the influence of the interaction. We find very small errors in the rate, not

tabulated here. The entry λo1 refers to the constant in formula (6.1), which

gives an offset for the width of the rarefaction wave. This entry is expressed in

mesh units. We form a linear model for this constant in Table 6.2. Similarly,

the compression wave width has the form (6.1). Here the rate has a negative

sign. We find very small errors in the rate, not tabulated here. The entry λo3

refers to the constant in formula (6.1), which gives an offset for the width of

the compression wave. This entry is expressed in mesh units. We form a linear

73

model for this constant in Table 6.2.



ωo1 (l. sonic) 0.020 -0.073 0.697 0.33% 0.0002

ωo2 (contact) 0.198 1.076 -0.716 0.30% 0.0006

ωo3 (r. sonic) -0.029 0.107 0.295 0.76% 0.0003

λo1 (l. sonic) 4.613 0.018 7.378 10.5% 0.206

λo2 (contact) -3.480 -0.514 19.137 11.8% 0.091

λo3 (r. sonic) 6.260 0.777 3.812 3.09% 0.111

po1 (l. sonic) 0.019 0.205 0.524 21.21% 0.023

po2 (contact) -0.12 0.023 0.779 0.783% 0.023

po3 (r. sonic) 0.186 0.056 0.410 25.23% 0.027

Table 6.2: Case 5. The SNRP defined by the contact rarefaction interaction.Expansion coefficients for output wave strengths (linear model) for input vari-ation ±10%.

Case 6: Reflected shocks from interactions 1 and 3 over-

take

In this case, the overtake of reflected shocks from interactions 1 and 3 is

studied. The left state is held fixed and the two wave strengths are varied.

Figure 6.6: Problem 6: Shock-shock overtake (two waves of the same family)

74

The two input shocks are both left moving forward shocks, which are

produced by interactions 1 and 3, respectively. One left moving forward shock,

one contact and one right moving backward shock are produced. We ignore

the output backward shock because it is too weak to be recognized by the

filter program. Therefore, the position error po3 and the width error λo

3 are not

presented here.

However, with an improved post-processing program, we can still get all

input and output gas states, so all input and output strengths are studied

here.

From the analysis of the section 2.2.1, the contact width will be modeled

as wc ∼ cct1/3, where the coefficient cc is affected to Mach number, and the

t denotes the time steps counted from the interaction time. Thus we model

this case in the same manner as the step down problem. This also is showed

by the Fig. 6.7, in which we plot the time step axis on a log scale to show

this relationship. The plot starts from 51, because the contact widths near

interaction time are unstable, and do not follow this scaling law. We observe

an increase in width from 4 cells to 9 over 1000 time steps.

In this plot there is a noticeable small disturbance after 1000 time steps,

beyond an otherwise good linear relation. That disturbance is due to an addi-

tional wave interaction with the contact at that time, which therefore affects

the contact width after 1000 time steps. This interaction is not presented here

since it is not one of the major interactions, but still it can be seen in Figure

10 in the paper [11].

Thus, in Table 6.3 we interpret λo2t

1/3 or λo2 as the output contact wave

75

Figure 6.7: Because the width is entirely grid related, we record width in unitsof ∆x and time in units of the number of time steps.

width, also in grid units, depending on the Mach number. For λo1, it is the

shock wave width and independent of time steps. And pak and ωa

k (k = 1, 2, 3

and a = i, o) represent position errors and wave strengths as usual.

The width of the left most input forward shock, which is from interaction

1, is 1 grid unit. The width of the other input forward shock, which is from

interaction 3, is 2 grid units.

The output position errors are computed assuming zero input position

error to see their relationship with the input wave strengths, according to the

formulae introduced by Section 2.3.2.

76


2 error(l. sonic) (l. sonic) L∞ STD

ωo1 (l. sonic) 0.033 0.375 1.110 0.02% 0.0001

ωo2 (contact) -0.028 0.075 0.149 0.91% 0.0002

ωo3 (r. sonic) -0.018 0.014 0.056 0.57% 0.0000

λo1 (l. sonic) 2.050 -0.248 -0.224 9.57% 0.078

λo2 (contact) 0.092 0.278 1.418 1.71% 0.007

po1 (l. sonic) 0.132 -0.030 0.007 12.92% 0.009

po2 (contact) -0.026 0.010 0.233 32.06% 0.009

Table 6.3: Case 6. The SNRP defined by the shock shock overtake (two wavesof the same family). Expansion coefficients for output wave strengths (linearmodel) for input variation ±10%.

Case 7: Compression wave reflected from interaction 5

reflects off of wall

Here we study the reflection of the compression off the wall, a special case

of the compression crossing compression interaction. The right state is held

fixed.

Figure 6.8: Problem 7: Compression-wall

Similar to the previous cases, the compression wave width has the form

(6.1). Here, the initial width denotes the compression width, which would

77

occur at the interaction time without the influence of the interaction. The rate

has a negative sign. and, we find very small errors in the rate, not tabulated

here. The entry λo1 refers to the constant in formula (6.1), which gives an

offset for the width of the compression wave. This entry is expressed in mesh

units. We form a linear model for this constant in Table 6.4. We find less

than 0.001% density oscillation near wall. So, there is no wall error in the

compression wall interaction.



1 (l. sonic) -0.026 1.127 1.450% 0.00061λo

1 (l. sonic) -0.893 -2.323 31.41% 0.107po

1 (l. sonic) 0.034 -0.223 2028% 0.026

Table 6.4: Case 7. The SNRP defined by the crossing of two compressions.Expansion coefficients for output wave strengths (linear model) for input vari-ation ±10%.

Case 8: Compression wave from wall reflection crosses

contact

After reflection from the wall, the transmitted compression wave re-

crosses the reflected contact. The right state velocity v = 0.0 is fixed and

the left state densities and pressures are held fixed.

We have two input wave width parameters, one each for the contact and

the compression, and three output wave width parameters, each for the left

compression, the contact and the right rarefaction wave. The entry λo1 in

78

Figure 6.9: Problem 8: Contact-compression

Table 6.5 refers to the stable width of the transmitted compression wave after

it is changed to a shock wave.



ωo1 (l. sonic) 0.030 -0.039 0.601 3.46% 0.0008

ωo2 (contact) -0.043 0.973 0.442 0.71% 0.0011

ωo3 (r. sonic) -0.026 0.034 0.402 4.29% 0.0007

λo1 (l. sonic) 3.298 0.116 -1.510 13.2% 0.080

λo2 (contact) -0.240 -0.081 -4.584 3480% 0.432

λo3 (r. sonic) 3.758 -1.265 9.379 19.7% 0.273

po1 (l. sonic) -0.311 1.842 -2.484 137.5% 0.263

po2 (contact) 0.228 -0.230 -0.081 239.4% 0.147

po3 (r. sonic) -0.440 -0.935 2.665 49.3% 0.166

Table 6.5: Case 8. The SNRP defined by the contact compression interaction.Expansion coefficients for output wave strengths (linear model) for input va-ration ±10%.

79

Case 9: Rarefaction reflected from interaction 8 reflects

off of wall

Case 9 is the reflection of the rarefaction off the wall again. We used

the same functional form for the error model as in case 4, but the numerical

coefficients are quite different. The right state velocity v = 0.0 is fixed and

the left state densities and pressures are held fixed.

Figure 6.10: Problem 9: Rarefaction-wall



1 (l. sonic) 0.004 0.938 0.55% 0.0001λo

1 (l. sonic) -0.060 5.290 206% 0.067po

1 (l. sonic) -0.027 -0.965 1133% 0.036

Table 6.6: Case 9. The SNRP defined by the crossing of two rarefactions.Expansion coefficients for output wave strengths (linear model) for input vari-ation ±10%.

80

Case 10: Rarefaction reflected off of wall passes through

contact

The output wave strengths in case 10 are very weak (ωo1 = 0.033, ωo

2 =

0.674, ωo3 = 0.021). The wave filter lacks sufficient precision to allow analysis

of the errors in this case, which because their small size are not important in

any case. The right state velocity v = 0.0 is fixed and the left state densities

and pressures are held fixed.

Figure 6.11: Problem 10: Contact-rarefaction

6.2 Errors in Resolved Calculations

We examine errors in wave strength, wave position and wave width, based

on the graphical expansion given in the Chapter 2. The wave strength errors

are dominated by the transmission of error (or uncertainty) from the initial

conditions. The wave width errors and the wave position errors are dominated

by the created error in the current interaction and the transmission of errors

from previous interactions. In Tables 6.7, 6.8, 6.9, 6.10, 6.11 and 6.12, we

compare the predicted error with the error computed directly, taken from a

81

full solution of the multiple wave interaction problem. The model for the

prediction of the error is satisfactory for those cases: the wave strength and

its errors, the wave width errors.


ωo1 (l. sonic) 0.346 0.348



λo1 (l. sonic) 4.568 4.474


1 (l. sonic) 0.356 0.244

Table 6.7: Case 4. The crossing of two rarefactions. Predicted and simulatederrors for output wave strengths, wave widths and wave positions.

82


ωo1 (l. sonic) 0.212 0.212

ωo2 (contact) 0.675 0.672

ωo3 (r. sonic) 0.147 0.145


1 (l. sonic) 0.0002 0.0002Var ωo

2 (contact) 0.0011 0.0014Var ωo


λo1 (l. sonic) 7.212 7.176

λo2 (contact) 2.417 2.790

λo3 (r. sonic) 7.895 8.101


1 (l. sonic) 1.334 0.941po

2 (contact) 0.021 0.010po

3 (r. sonic) -0.159 -0.042

Table 6.8: Case 5. The contact rarefaction interaction. Predicted and simu-lated errors for output wave strengths, wave widths and wave positions.

83


ωo1 (l. sonic) 0.783 0.781

ωo2 (contact) 0.079 0.083

ωo3 (r. sonic) 0.015 0.017


1 (l. sonic) 0.0016 0.0019Var ωo

2 (contact) 0.000001 0.000005Var ωo


λo1 (l. sonic) 1.808 1.821

λo2 (contact) 3.947 4.248


1 (l. sonic) 0.678 0.427po

2 (contact) 0.797 0.589

Table 6.9: Case 6. The shock shock overtake. Predicted and simulated errorsfor output wave strengths, wave widths and wave positions.


ωo1 (l. sonic) 0.138 0.139



λo1 (l. sonic) 8.183 6.662


1 (l. sonic) 0.152 0.038

Table 6.10: Case 7. The crossing of two compressions. Predicted and simulatederrors for output wave strengths, wave widths and wave positions.

84


ωo1 (l. sonic) 0.086 0.086

ωo2 (contact) 0.675 0.674

ωo3 (r. sonic) 0.053 0.053


1 (l. sonic) 0.00003 0.00003Var ωo

2 (contact) 0.0013 0.0013Var ωo


λo1 (l. sonic) 2.843 3.168

λo2 (contact) -1.356 -0.927

λo3 (r. sonic) 12.794 11.381


1 (l. sonic) 1.321 0.613po

2 (contact) 0.186 0.078po

3 (r. sonic) -1.526 -0.715

Table 6.11: Case 8. The contact compression interaction. Predicted andsimulated errors for output wave strengths, wave widths and wave positions.


ωo1 (l. sonic) 0.054 0.053



λo1 (l. sonic) 11.293 10.013


1 (l. sonic) 1.954 0.669

Table 6.12: Case 9. The crossing of two rarefactions. Predicted and simulatederrors for output wave strengths, wave widths and wave positions.

85

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