imac xxviimodels numerical solution error using time series

7/27/2019 IMAC XXVIIModels Numerical Solution Error Using Time Series

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Identifying Models of Numerical SolutionError Using Time Series Analysis

Franois M. Hemez1 Marine Marcilhac2

Applied Physics Division (X-Division)Los Alamos National Laboratory

Los Alamos, New Mexico 87545, U.S.A.

Communication and Systems (C-S)22 Avenue Galile

92350 Le Plessis Robinson, France

ABSTRACT:The main goal of this research is to identify mathematical models that describe thebehavior of truncation error for arbitrary systems of partial differential equations solved bynumerical methods. Methods of interest include finite element, finite difference, finite volumeand particle methods. Truncation error is the difference between the exact solution ofcontinuous equations and the numerical solution of discretized equations. Understanding howtruncation error behaves and describing it with a mathematical model are essential steps ofcode verification activities. We propose to apply a MATLAB-based time series analysis toolbox

developed at the University of Lancaster, U.K., and known as CAPTAIN, to numerical solutionsobtained by running a computer code. The analysis is framed such that the CAPTAIN toolboxidentifies a transfer function between the discrete solution and its truncation error. Onceidentified, the transfer function is converted to an ordinary differential equation that describesthe numerical error and, in fact, represents the (possibly unknown) modified equation of thealgorithm. A proof-of-concept is provided using datasets generated with a linear, harmonicoscillator and a non-linear ordinary differential equation. The technique proposed performs wellon these examples and a path forward is proposed to take it one step further and identify therate-of-convergence of a numerical algorithm using mesh refinement.

1. INTRODUCTION

Code and solution verification are defined as a scientifically rigorous and quantitativeprocess for assessing the mathematical consistency between continuum and discretevariants of partial dif ferential equations used to represent a reality of i nterest, as stated inReference [1]. Verifying the numerical accuracy of discrete solutions computed by simulationcode is important because partial differential equations that govern the equations of motion orconservation laws in computational engineering and physics are discretized for resolution withfinite-digit arithmetic. The main challenge of assessing the performance of an algorithm and,therefore, the numerical quality of discrete solutions it delivers is to understand the extent towhich solutions of the discretized equations converge to the (often unknown) exact solution ofthe continuous equations. Other Verification and Validation (V&V) activities used to establish thelevel of confidence that can be placed in a Modeling and Simulation (M&S) capability are

discussed in Reference [2].

1Technical staff member in X-Division. Mailing: Los Alamos National Laboratory, X-3, Mail Stop B259, Los Alamos,

NM 87545, U.S.A. Phone: (+1) 505-667-4631. E-mail: [email protected] and development engineer at C-S. Formerly post-doctoral research assistant in X-Division, Los Alamos

National Laboratory. Mailing: Communication and Systems, C-S, 22 Avenue Galile, 92350 Le Plessis Robinson,France. Phone: (+33) (0)-1-41-28-47-82. E-mail: [email protected].

Proceedings of the IMAC-XXVIIFebruary 9-12, 2009 Orlando, Florida USA

2009 Society for Experimental Mechanics Inc.
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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In short, code verificationis the assessment that the computer code does what it is supposedto do for an application of interest. It includes activities such as software quality assurance,verifying that there is no programming deficiency, no significant round-off or truncation errors,etc. The activities of solution verification, on the other hand, focus on demonstrating that thecomputational discretization provides adequately converged solutions. Solution verification andself-convergence study the error originating from the reliance on finite arithmetic and predict thenumerical error (or solution error) obtained from running the calculation using specific values of

mesh or grid sizes x, time steps t, etc. These activities are discussed in Reference [3].A thread common to the above activities, whether one deals with code or solution verification, isto assess how truncation error behaves. This is because numerical methods always provideapproximations. Finite element methods, for example, solve weak formulations of the equationsof motion. The theory then states that weak and strong solutions become equivalent as x0.They are not, however, equal and the difference between the two is captured by truncation errorwithin the domain of asymptotic convergence.

The procedure most commonly encountered to assess the behavior of truncation error is therefinement of a computational mesh. A discussion is provided in Reference [4]. One difficulty,however, is that the functional formof the mathematical model that one seeks to identify fortruncation error is often unknown. It is common practice to assumethe form of truncation error

using a low-order polynomial such as, for example, y

Exact

y(x) + .(x

p

) where y

Exact

denotesthe exact solution of the continuous equations, y(x) is the discrete solution obtained on thebasis of discretization x and the last term .(xp) represents truncation error. Clearly, theresults of a verification study are vulnerable to the correctness of this critical assumption.

We propose a novel approach toassess the leading-order term of truncation error withou thaving to assume a priori what its functional form may be. The technique identified modelsof transfer functions between discrete solutions computed by a code and their truncation errors.These transfer functions can then be converted into ordinary differential equations that governhow truncation error behaves. The technique proposed can be applied to any set of equations ofmotion or conservation laws that are solved with an arbitrary numerical method.

2. THE TRUCATION ERROR OF A NUMERICAL METHOD

The accuracy of a numerical method is, by definition, the difference between the exactsolution of the continuous equations and discrete approximation estimated using a mesh or gridsize x. In the following, the exact solution is referred to as yExact and the discrete solution isdenoted by the symbol y(x) to stress the fact that it results from discretization x. Solutionerror is defined simply as:

( ) x)y(yx Exact = . (1)

The implementation of a numerical method, such as a finite element method, finite volumemethod or finite differencing scheme, always involves approximations whereby representationor basis functions are truncated for coding on finite-arithmetic architectures. Clearly the degreeto which the numerical solution is an accurate approximation of the exact solution depends onthe order of truncation of the method. Assuming that truncation error is the dominant source oferror, this can be represented mathematically as:

( ) ( )1ppExact xOx.x)y(y +++= , (2)where the term .(xp) represents the leading order of truncation error. The symbol p denotesthe observed rate-of-convergence of the numerical method. For example, a problem that usesquadratic finite elements should recover p = 2. Comparing equations (1) and (2) results in:


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( )px.(x) . (3)

It is emphasized that equality (3) assumes that truncation error is the most significant source ofnumerical error. Other potential sources of error include the lack-of-convergence of an iterativesolver, numerical round-off and inappropriate resolution, to name only a few. The aboveassumption is precisely what defines the regime of asymptotic convergence. It is the range ofresolution levels, xMinx xMax, within which truncation error dominates all other sources oferror. An example of asymptotic convergence is illustrated in Figure 1. It pictures the behavior oferror (1) when the Burgers equation is solved with a Lax-Wendroff integration scheme. The levelof spatial resolution of the calculation is varied within 103x 1 (cm) and clear evidence offirst-order convergence is observed, as indicated by the slope p 1.2 of the curve on Figure 1.First-order convergence matches the theoretical order of truncation of the numerical scheme forthese calculations that feature discontinuous solutions.

p = 1

Figure 1. Illustration of asymptotic convergence for the 1D, Burgers equation.

A numerical method and the discrete solutions it provides derive their accuracy from the factthat truncation error is under control. Being able to derive a mathematical model of truncationerror is at the heart of verification activities. Such models are needed to verify the order of

convergence of a numerical method, extrapolate discrete solutions as x

0 and quantifysolution uncertainty. Common practice is, unfortunately, to assume how truncation errorbehaves and postulate its mathematical form. This practice can lead to grave mistakes, shouldassumptions made about the mathematical function used to represent the error be incorrect.This work attempts to address this issue by proposing a technique that treats the analysis codeas a black box and can inform on an appropriate choice of model for a particular simulation ornumerical application.


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3. IDENTIFYING A MODIFIED EQUATION THROUGH TIME SERIES ANALYSIS

Other than the technique known as Modified Equation Analysis (MEA), there is no methodto systematically derive mathematical models of truncation error. MEA is a general-purposetechnique capable of exactly identifying the mathematical form of truncation error. Overviewsare provided in References [5-6]. But its applicability is limited to relatively simple equations andnumerical scheme. The derivations rapidly become intractable when applied, for example, tosystems of finite element or finite volume equations, even with the help of a symbolic solver. It is

to address this limitation that an alternative technique based on signal processing is proposed.

The approach that we propose takes advantage of the duality between transfer functions and(linear) ordinary differential equations. The procedure is articulated in two steps. The first step isto carefully select input and output responses from a numerical simulation and identify atransfer function that relates them. The second step is to convert this transfer function into anordinary differential equation. If these inputs and outputs represent the discrete solution andits truncation error, we will have obtained a differential equation that connects them, which isnothing less than the modified equation that defines truncation error.

The theoretical framework is summarized below. A linear, differential equation that relates aninput function F(x) to an output function y(x) can be generically written as:

(x)xF(x)

xFF(x)(x)

xy(x)

xyy(x) m

m

m10n

n

n10+++=+++

LL , (4)

where coefficients 0 nand 0 mare constant. Note that the symbol x of equation (4)does not necessarily represent a spatial dimension. It could, for example, mean time or energy.Using a Laplace transform, the differential equation can be converted to:

y(s)H(s)F(s)= , (5)

where y(s) and F(s) are the Laplace, or frequency-domain, transforms of functions y(x) and F(x),respectively. The transfer function is defined as:

m

m10

n

n10

ss

ssH(s)

+++

+++=

L

L. (6)

We propose to rely on the theory represented by equations (4-6) but to proceed backwards.The transfer function is, first, identified from the appropriate responses. It is used next to providethe leading-order term of the differential equation that controls how truncation error behaves.

The implementation of a pth-order accurate numerical method defines a truncation error whoseleading order takes, in general, the form:

( ) t)(x;x

y.x.t)(x;

q

qp

, (7)

where is a function that depends on properties of the system of partial differential equationssolved, but that is independent of spatial discretization x. Solving, for example, a structuralmechanics problem with a quadratic finite element method yields p = 2 and q = 3. Equation (7)suggests how the input F(x) and output y(x) can be selected to simplify the identification. Ifthe input is a discrete solution provided by the code and the output is the correspondingtruncation error, then the transform of equation (7) in the frequency domain becomes:

( ) y(s).ss(s)H(s)

n

n10444 3444 21

L+++= , (8)

where coefficients 0, 1 ndepend on the type of numerical method and discretization x.


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By selecting the input and output responses as the discrete solution and its truncation error,

Figure 2. Flow-chart of the method proposed to identify modified equations.The im series

on-linear, input-

the model of transfer functions is reduced to the numerator of equation (6), which implies fewercoefficients to identify. A flow-chart that defines this approach in the context of code verificationis shown in Figure 2.

plementation of Figure 2 relies on a methodology, and associated toolbox, for timeanalysis developed for over two decades at the University of Lancaster, U.K. The software iscalled CAPTAIN and comes in the form of a MATLAB-based toolbox that operates on arbitraryresponses. The identification decomposes these responses into dominant modes, as explainedin References [7-8], then, best-fits the generic model of transfer functions shown in equation (6).Given the orders (n; m) of equation (6) that must be specified by the user, in addition to a fewother settings, the toolbox identifies the best possible transfer function. Statistics of goodness-of-fit are estimated, which assists the user in determining the quality of the model identified.Details about the toolbox and its application can be found in References [9-10].

The CAPTAIN toolbox is fed with responses that can originate from linear or n

output systems. No particular knowledge is needed about the underlying system that generatesthese signals. The fact that the dynamic system is treated as a black box is appealing whenconsidering the application of the toolbox to code and solution verification. It is noted that, in ourapplication, one need not restrict the analysis to time series. Responses analyzed can representsimulation results defined as a function of time, space or any other discretization variable. Theonly limitation is that the procedure illustrated in Figure 2 can only be applied to test problemsfor which an exact solution of the continuous equations is known.

Perform NumericalSimulations with

Different (x; t)

Exact Solution

Code

System Identification

Estimate the Solution Error,

(x;t) = yExact y(x;t)

Estimate the Transfer Function,(s) = H(s).y(s)

Toolbox

Inverse Laplace

Transform

Estimate the Ordinary Differential

Equation of Truncation Error


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4. PROOF-OF-CONCEPT WITH A DAMPED, HARMONIC OSCILLATOR

A proof-of-concept is first discussed. The approach proposed is applied to the resolution ofa linear, single degree-of-freedom (SDOF) harmonic oscillator with a first-order accurate finite

ided to verifyhowdifferencing method. The Euler forward algorithm is used. This application is prov

the method performs when everything (exact solution and modified equation) is known.

MEA demonstrates that applying the Euler forward integration scheme to the harmonic oscillator

solves the following ordinary differential equation more accurately that the original equation:

)(tt

tf(t)y(t)(t)t

(t)t

(t)

22

43421

=+

+

, (9)yyy 22

meaning that the differential equation that governs truncation error is given by:

(t)t

yt(t)

2

2

= . (10)

Next equation (10) is discretized with the Euler forward time integration scheme implemented tosolve the original harmonic oscillator. Using the same scheme is important for consistency. Thediscrete version of equation (10) is:

+++=

t

y[k]1]y[k22]y[k[k] . (11)

Using a control theory notation that mixes the sampled, time-domain quantities y[k] and [k] andcontinuous transfer function H(s), equation (11) can be described symbolically as:

( )y[k]ss21t

[k]

H(s)

2

44 344 21

+= . (12)

It is emphasized that the notation of equation (12) is symbolic. It is meant to represent the filterH(s) that acts on sampled inputs y[k] to produce outputs k]. The notation would not, clearly, beused for an implementation since it confuses time-domain and frequency-domain quantities.

3)

[

The point this derivation is to show that we are looking for a model of transfer function that canbe expressed as a second-order polynomial:

2ssH(s) ++= , (1210

and, because the modified equation is derived explicitly in equation (12), the triplet of polynomialcoefficients ( ; ; ) is also known:0 1 2

t0 = ,

01 2

t == and

202

t == . (14)

In general the transfer function and its coefficients are, of course, unknown. The user would,then, search for the polynomial form and coefficient values that best-fit the available data. The

fact that the exact solution is, known is used to our advantage to verify that the procedure

ion is known, the solution error (t) = yExact y(t) is computed for each

here,yields sensible results.

The numerical application presented next integrates the harmonic oscillator over the time period0 t 2 sec. Three solutions are obtained with time samples t = 4.103, 103and 2.5 104sec.Because the exact solutresponse. The CAPTAIN toolbox is then provided with pairs of time series (y(t); (t)) for eachone of the three runs. Figures 3 and 4 show the discrete solutions and truncation errors when


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analysis is performed with time steps t = 4 milli-sec. (in Figure 3) and t = 0.25 milli-sec. (inFigure 4). The figures also compare the original truncation errors (dashed, red lines) to thereconstruction provided by CAPTAIN (green, solid lines). The identification of transfer functionsleads to reasonable goodness-of-fit to data, both in terms of amplitude and phase.

Time Samples

Time Samples

DiscreteSolution,

y(t)

ation

(t)

Trunc

Error,

Figure 3. Identification of truncation error at t = 4.10 sec. for the linear oscillator.(Top: Comparison of truncation errors (t); bottom: discrete solution y t).)

3

(

TruncationError, (t)

Time Samples

DiscreteSolution,

y(t)

Time Samples

Figure 4. Identification of truncation error at t = 2.5.104sec. for the linear oscillator.(Top: Comparison of truncation errors (t); bottom: discrete solution y(t).)


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Table 1 summarizes the results of fitting transfer function models (13) to the datasets. Values ofthe triplets of identified coefficients (0; 1; 2) are listed, together with goodness-of-fit statisticsestimated by CAPTAIN. Table 2 scales coefficients 0, 1and 2according to equation (14) toeliminate their dependence on time step t, which makes it easier to compare them. It can beobserved that the identification is consistent in the results it provides, as one would expect.

Table 1. Identification results for the linear oscillator.

Discrete Solution, y(t)Quantityof Interest

Coarse Medium Fine

Time Step t 4.103sec. 103sec. 2.5.104sec.

Identified Coefficient 0 94.7 380.6 1,458

Identified Coefficient 1 -187.8 -754.2 -2,889

Identified Coefficient 2 93.1 373.6 1,430

Goodness-of-fit Statistic 78.4% 87.4% 97.4%(Legend: The goodness-of-fit statistic is an indicator that is scaled in the 0-to-1 range.)

Table 2. Scaled coefficients kIdentified with CAPTAIN for the linear oscillator.

Discrete Solution, y(t)Quantity

of Interest Coarse Medium Fine

Corresponding Coefficient ( t) 0.379 0.381 0.3650Corresponding Coefficient (-1t/2) 0.376 0.377 0.361

Corresponding Coefficient (2t) 0.372 0.374 0.357

The overall observation we draw from this first example is that the method performs accordingto expectation. Given sampled datasets, it is capable of identifying a transfer function that canbe converted back to an ordinary differential equation of the modified equation. But this exampleonly provides a crude sanity check that the approach is reasonable. A more complicated datasetis analyzed next to

5. APPLICATION TO THE NON-LINEAR, HYPERBOLIC BURG RS EQUATIONvel up in complexity is the an on-lin The w

B s mathemata tinuous solutions, given appropriate initia tions. After giving ab blem and its modified equation, the results of a numerical simulationp r method are discussed

5

T non-linear and hyperbolic and bears resemblance with other systems of equations, such as the Eulerequations of gas dynamics or Navier-Stokes equations of computational fluid dynamics, in its

ability to deve uous solutions. Th ous Burgers equation is defin

assess the performance of the method.

EearThe next le alysis of a n

ical properties (it is a hyperbolic equation) andequation. ell-known, 1D

urgers equation is selected for itbility to generate disconrief summary of the pro

l condi

erformed with a first-orde .

.1 Description of the Burgers Equation and its Modified Equation

he equation chosen for analysis is Burgers equation in 1D, Cartesian geometry. It is

lop discontin e continu ed as:

t)(x;x

yt); =(x

x

y

2

12

22

, (15)t)(x; +

t

y

where a diffusive term is added in the right-hand side. Coefficient is a user-defined constant. Afirst-order, upwind method is used to integrate equation (15). Details of this numerical scheme


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are discussed in Reference [11]. The upwind method, also known as Euler backward in othercircles, is one of the standard techniques used in computational engineering and physics.

Even tough equation (15) and the first-order upwind method are both simple, the derivationsneeded to carry out the MEA become intractable by hand. Using, however, the symbolic solver

elow:MathematicaTM, a solution can be obtained. The leading term of truncation error is shown b

44444444444 344444444444 21

L

t)(x;

4

4

2

2

3

32

2

2

2

22

x

y

6

x

y

x

y

2

1

x

yy

6

1x

x

yx

2

y

x

y

x

y

2

1

t

y+

+

+

=

+

,

(16)

where the notation y = y(x;t) is used for simplicity. Equation (16) indicates that the numericalmethod applied to Burgers equation is first-order in space. A similar analysis leads to second-

s defined as:order in time (not shown). The dominant term of spatial truncation error i

t)(x;x

yx

2

t)y(x;t)(x;

2

2

= . (17)

Because the highest-degree derivative in equation (17) is 2nd-order, one can search for atransfer function H(s) = 0+ 1s + 2s

2 that assumes the form of a 2nd-order polynomial.advantage of analyzing this problem is that, again, the triplet of coefficients (0; 1; 2) is knownexactly and can be shown to be equal to:

One

x2

y0 = , 01 2

x

y == and 02

x2

y == . (18)

The values of (0; 1; 2) identified with CAPTAIN can be compared to these exact solutions to

5.2 Identification of the Modified Equation for Burgers Equation

A total of four discrete solutions are computed over the spatial domain 0 x cmsuccessively refined meshes. The coarsest level of mesh resolution is a uniform grid sampled at

x = 4.102

cm. This first mesh defines a total of 25 grid points at which the solution y(x;t) is

Table 3. Identification results for the non-linear Burgers equation (upwind method).

assess the performance of the method when datasets are, here, generated from the analysis ofa non-linear equation.

1 using

evaluated. A refinement ratio of R = 2 is used, meaning that the grid is halved each time togenerate the other three computational meshes. The Burgers equation is integrated to the finaltime of t = 1 sec., starting from a smooth initial condition. (It produces no discontinuity.)

Discrete Solution, y(x;t)Quantityof Interest

Extra-coarse Coarse Medium Fine

Mesh Size x 4.102cm 2.102cm 102cm 5.103cm

Number of Grid Points 25 50 100 200

Time Step t 103 3 3 3sec. 10 sec. 10 sec. 10 sec.

Number of Time Samples 1,000 1,000 1,000 1,000Identified Coefficient 0 12.93 26.05 55.15 117.40

Identified Coefficient -28.02 -54.37 -112.51 -236.891Identified Coefficient 2 14.60 27.65 56.64 118.76

Goodness-of-fit Statistic 85.4% 94.8% 98.1% 99.4%(Legend: The goodness-of-fit statistic is an indicator that is scaled in the 0-to-1 range.)


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Table 3 summarizes the results of fitting transfer functions H(s) = 0+ 1s + 2s2to the datasets.

In Table 4, the regression coefficients 0, 1and 2are scaled to eliminate their dependence onmesh size x and facilitate the comparison to the exact solution.

Table 4. Scaled coefficients kIdentified with CAPTAIN for Burgers equation.

Discrete Solution, y(t)Quantityof Interest

Extra-Coarse Coarse Medium Fine

Corresponding Coefficient (0x) 0.517 0.521 0.551 0.587

Corresponding Coefficient (-1x/2) 0.560 63 0.5920.544 0.5

Corresponding Coefficient (2x) 0.584 66 0.5940.553 0.5

Exact S 00 .500 0olution (|y|/2) 0.5 0 0.50 0.500

For this numerical application, the exact value of th that a a verequation (18) is equal to ; it defines a reference for comparison with the identified co .C ; ) listed in Table asona ent w ct vab 0-to-20% erro eir values able while the identified transferf of fitting the pa iscrete solution; truncation error) respona indicated by th oodness atisticst f robustness in ethod is welcome because lies thc ntification of the coefficients. Even if they are present, these errors don a

e term |y|/2 ppe rs se al times inefficlue. It can

ientsoefficients (0; 1 2e further observed that a 1

4 are in re ble agreem ith the exar in th is toler

unction remains capable ir of (d ses withcceptable accuracy, as e high g -of-fit st of Table 3. What seemso be an inherent level o the m it imp at errorsan be tolerated in the ideot jeopardize the identification of the leading term(s) of the modified equ tion.

Based on results of Table 3, one would conclude that the transfer function between a discretesolution and its truncation error is a 2nd-order polynomial. It corresponds to a truncation errorwhose dominant contribution obeys an ordinary differential equation of the type:

( ) t)(x;y

.t)(x;2

2 x .

x(19)

for Burgers equation discretized with the upwind method. The identified hthe dominant ial truncation e oef x) o (19) dso defines r of acc f the n al me he

r nown point an proac gested next toi ment stu

s of the transfer

This is illustrated, for example, in equation (19) that features a coefficient (x) that depenthe mesh size. Performing, say, three runs with coarse ( C), medium (xM) and fine (xF)

model (19) agrees witterm of spat rror (17). The c ficient ( f equation depen

n the mesh size and this function the orde uracy o umeric thod. T

ate-of-convergence is, however, unk at this d an ap h is sugdentify its value from a mesh refine dy.

6. A PATH FORWARD TO IDENTIFY THE ORDER OF ACCURACY

As explained above, we have so far focused on the identification of a transfer function thatrepresents an unknown, dynamic system between discrete solutions and their truncation errors.The transfer function can, then, be converted into an ordinary differential equation that modelsthe leading-order term of a modified equation. This procedure, however, says nothing about theformal order of accuracy (or rate-of-convergence) of truncation error.

In Reference [12], it is proposed to augment the approach illustrated in Figure 2 by adding theestimation of the order of accuracy of the numerical method. To do so, multiple calculations are

needed from a mesh refinement study. These runs provide several estimationfunction coefficients. Because these coefficients are estimated from calculations performed withdifferent mesh sizes (or time steps), a relationship can be established between their values andthe rate-of-convergence of the numerical method.

ds onmeshx


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sizes yields a triplet of leading-order coefficients {(xC); (xM); (xF)} from which a power-lawmodel such as (x) = .(xq) can be identified. The exponent q is directly related to the orderof accuracy. Results obtained in Reference [12] indicate that this approach recovers rates-of-convergence that are in excellent agreement with the formal order of accuracy of the method.

7. CONCLUSION

ation error withouthavi

e University of Lancaster,

h periodicity, linearity or stationarity are not

redevelopment could include application to coupled physics test problems where one algorithm

tly from another one in terms of truncation error. One example would be aoblem where a 1st-order finite element method is used to calculate the

mec

dvanced Scientific Computing (ASC)

tion, Calculation Verification,

A novel approach is proposed to assess the leading-order term of truncng to assume a priori what its functional form may be. The technique is based on identifyingmathematical models of transfer functions between discrete solutions computed by a code andtheir truncation errors. These transfer functions can then be converted into ordinary differentialequations that govern how truncation error behaves. The identification is carried out using aMATLAB-based toolbox for time series forecasting developed at thU.K., that decomposes time series into dominant modes.

The approach can assess the dominant effect of truncation error for any system of conservationlaws that are solved with an arbitrary numerical method. The computer code is treated as ablack box. It is used simply to obtain discrete solutions from a mesh refinement study. Discretesolutions and truncation errors provided to the forecasting toolbox do not necessarily need to besignals defined in time. Likewise assumptions suc

needed. These attributes make it attractive to apply the technique to a diversity of algorithms incomputational engineering and physics. The main limitation is that an exact solution of thecontinuous equations is needed to estimate truncation error, which restricts applicability of themethod to code verification test problems.

The examples discussed in this report are simple proofs-of-concept. More work is warranted toapply the technique to other test problems and gain confidence in its performance. Futu

may behave differenstructural-thermal pr

hanical stresses while a 2nd-order finite volume method is applied to the thermal field. Eventhough the modified equation of either algorithm may be well characterized, truncation error forthe coupled calculation may behave in a manner that is different from the sum of its parts.

ACKNOWLEDGMENTS

This work is performed under the auspices of the AVerification and Validation (V&V) program at Los Alamos National Laboratory (LANL). The firstauthor is grateful to Mark C. Anderson, V&V program leader, for his continued support. Also, thecontribution to this research of our colleagues Jerry S. Brock and James R. Kamm is recognizedand greatly appreciated. LANL is operated by the Los Alamos National Security, LLC for theNational Nuclear Security Administration of the U.S. Department of Energy under contract DE-

AC52-06NA25396.

BIBLIOGRAPHICAL REFERENCES[1] Brock, J.S., ASC Level-2 Milestone Plan: Code Verifica

Solution-error Analysis and Test-problem Development for LANL Physics SimulationCodes, Technical Report LA-UR-05-4212 of the ASC Code Verification Project, Los

Alamos National Laboratory, Los Alamos, New Mexico, May 2005.


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[2] Hemez, F.M., Doebling, S.W., Anderson, M.C., A Brief Tutorial on Verification andValidation, 22nd SEM International Modal Analysis Conference, Dearborn, Michigan,January 26-29, 2004.

[3] Roache, P.J., Verification in Computational Science and Engineering, Hermosaue, New Mexico, 1998.

[4]

F., Hyett, B.J., The Modified Equation Approach to the Stability andAccuracy Analysis of Finite Difference Methods, Journal of Computational Physics, Vol.

pp. 113-129.

ary 2007. Web resource: http://www.es.lancs.ac.uk/cres/captain

Publishers, Albuquerq

Hemez, F.M., Challenges to the State-of-the-practice of Solution Convergence

Verification, 9

th

AIAA Non-deterministic Approaches Conference, Oahu, Hawaii, April23-26, 2007.

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[6] Warming, R.

14, 1974, pp. 159-179.

[7] Young, P.C., Dominant Mode Analysis: Simplicity Out of Complexity, Proceedings of theSensitivity Analysis and Model Output (SAMO) Summer School, Venice, Italy, July 2006.

[8] Young, P.C., Data-based Mechanistic Modelling, Generalised Sensitivity and DominantMode Analysis, Computer Physics Communication, Vol. 117, 1999,

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