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CFD 6 1 David Apsley

6. TIME-DEPENDENT METHODS SPRING 20106.1 The time-dependent scalar-transport equation

6.2 One-step methods

6.3 Multi-step methods

6.4 Uses of time-marching in CFDSummary

Examples

6.1 The Time-Dependent Scalar-Transport Equation

The time-dependent scalar-transport equation for an arbitrary control volume is

sourcefluxnetamountt

=+)(d

d(1)

where: amount total quantity in a cell = mass concentration = V;flux is the rate of transport through the boundary .

In Section 4 it was shown how the flux and source terms could be discretised as

P

F

FFPP baasourcefluxnet = (2)

In this Section the time derivative will also be discretised.

We first examine numerical methods for the first-order differential equation

Ft=

d

d, 0)0( = (3)

where Fis an arbitrary function oftand . Then we extend the methods to CFD.

Initial-value problems of the form (3) are solved by time-

marching. There are two main types of method:

one-step methods: use the value from the previous timelevel only;

multi-step methods: use values from several previous times.

t

old

new

t told new

t

t

t(n+1)

(n+1)

(n)

(n-1)

(n-2)

t(n)

t(n-1)

t(n-2)

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6.2 One-Step Methods

For the first-order differential equation

Ft=

d

d(3)

the one-step problem is:given at timet(n) compute at timet(n+1)

The following notation is used:

identify everything at t(n) by a superscript old this is what we currently know; identify everything at t(n+1) by a superscript new this is what we seek.

By integration of (3), or from the figure above,

tFav= (4)

or

tFavoldnew += (5)These are exact. However, since the average derivative, F

av, isnt known until the solution

is known, it must be estimated.

6.2.1 Simple Estimate of Derivative

This is the commonest class of time-stepping scheme in general-purpose CFD. There are

three obvious methods of making a single estimate of the average derivative.

Forward Differencing(Euler Method)

Take Fav

as the derivative at the

startof the time-step:

Backward Differencing(Backward Euler)Take F

avas the derivative at

the endof the time-step:

Centred Differencing(Crank-Nicolson)

Take Fav

as the average of

derivatives at the beginning and end.

tFoldoldnew += tFnewoldnew += tFF newoldoldnew )(21 ++=

t

old

new

t told new t

old

new

t told new t

old

new

t told new

t12 t

12

For:

Easy to implement becauseexplicit(the RHS is known).

For:

In CFD, no time-steprestrictions;

For:

Second-order accurate in t.

Against:

Only first-order in t;

In CFD, stability imposes

time-step restrictions.

Against:

only first-order in t;

implicit (although, in CFD,

no more so than steady case).

Against:

Implicit;

In CFD, stability imposes time-

step restrictions.

t

old

new

t told new

t

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Classroom Example 1

The following differential equation is to be solved on the interval [0,1]:

1)0(,d

d==

t

t

Solve this numerically, with a step size t= 0.2 using:

(a) forward differencing;

(b) backward differencing;

(c) Crank-Nicolson.

Solve the equation analytically and compare with the numerical approximations.

Classroom Example 2

Solve, numerically, the equation

33

d

d

=

tt , 1)0( = in the interval 0 t 1 using a timestep t= 0.25 by:(a) the forward differencing (fully-explicit) method;

(b) the backward-differencing (fully-implicit) method;

(c) the Crank-Nicolson (semi-implicit) method.

Note that, since the equation is non-linear, the implicit methods (parts (b) and (c)) will

require iteration at every timestep.

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6.2.2 Other Methods

For equations of the form Ft=

d

d, improved solutions may be obtained by making

successive estimates of the average gradient. Important examples include:

Modified Euler Method(2 function evaluations; similar to Crank-Nicolson, but explicit)

)(

),(

),(

2121

12

1

+=

++=

=

ttFt

tFt

oldold

oldold

Runge-Kutta (4 function evaluations)

)22(

),(),(

),(

),(

432161

34

2

1

22

1

3

21

121

2

1

+++=

++=++=

++=

=

ttFtttFt

ttFt

tFt

oldold

oldold

oldold

oldold

More details of these and other advanced methods can be found in the course notes for the

Computational Mechanics unit.

For scalar , such methods are popular. Runge-Kutta is probably the most widely-usedmethod in engineering. However, in CFD, and F represent vectors of nodal values, andcalculating the derivative F (evaluating flux and source terms) is very expensive. The

majority of CFD calculations are performed with the simpler methods of 6.2.1.

Exercise. Using Microsoft Excel (or other computational tool of your choice) solve the

Classroom Examples from the previous subsection using Modified-Euler or Runge-Kutta

methods.

6.2.3 One-Step Methods in CFD

General scalar-transport equation:

0)(d

d

=+ sourcefluxnetVtP (6)

For one-step methods the time derivative is always discretised as

t

VVV

t

old

P

new

PP

)()()(

d

d (7)

Flux and source terms could be discretised at any particular time level as

PFFPP baasourcefluxnet = (8)Different time-marching schemes arise from the time level at which (8) is evaluated.

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Forward Differencing

[ ] 0)()( =+ old

PFFPP

old

P

new

P baat

VV

Rearranging, and dropping any new superscripts as tacitly understood:old

FFPPPP abat

Vt

V

++= )( (9)

Assessment.

Explicit; no simultaneous equations to be solved.

Timestep restrictions; for stability a positive coefficient of oldp requires 0 Pat

V.

Backward Differencing

[ ] 0)()( =+ new

PFFPP

oldPnewP baat

VV

Rearranging, and dropping any new superscripts:

oldPPFFPP

t

Vbaa

t

V)()(

+=+ (10)

Assessment.

Straightforward to implement; amounts to a simple change of coefficients:

old

PPPPt

Vbb

t

Vaa )(++ (11)

No timestep restrictions.

Crank-Nicolson

[ ] [ ] 0)()(21

21 =++

new

PFFPP

old

PFFPP

old

P

new

P baabaat

VV

Rearranging, and dropping any new superscripts:old

FFPPPPFFPP abat

Vbaa

t

V

+++=+ )()()( 2121212121

or, multiplying by 2 for convenience: old

FFPPPPFFPP abat

Vbaa

t

V

+++=+ )()2()2( (12)

Assessment.

Fairly straightforward to implement; amounts to a change of coefficients:old

FFPPPPPPP abat

Vbb

t

Vaa

++++ )()2(,2 (13)

Timestep restrictions; for stability, a positive coefficient of oldp requires 02 Pa

t

V.

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In general, weightings 1 and can be applied to derivatives at each end of the time step:

oldnew

P FFVt

)1()(d

d+ (14)

This includes the special cases of Forward Differencing ( = 0), Backward Differencing

( = 1) and Crank-Nicolson ( = ).

For 0 this so-called method can be implemented by a simple change in matrix

coefficients. For 1 (i.e. anything other than fully-implicit Backward-Differencing),

stability imposes a timestep restriction

old

Pa

Vt )(

1

1

< (15)

Example. Consider a 1-d time-dependent advection-diffusion problem

Sx

uxt

=

+

)()(

with the first-order upwind advection scheme on a uniform grid of spacing x, it is readily

shown that at any time-level the coefficients in the flux-source discretisation are

CDaaaDaCDa EWPEW +=+==+= 2,,

where the mass flow rate Cand diffusive transfer coefficientD are given by

x

ADuAC , ==

In the 1-d case the cross-sectional areaA = 1 and the time-step restriction (15) becomes

1

1

)(

2

2

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6.3 Multi-Step Methods

One-step methods use only information from time level t(n)

to

calculate (d/dt)av.

Multi-step methods use the values of at earlier time levels aswell: t

(n-1), t

(n-2), ... .

One example is Gears method:

tt

nnnn

2

43

d

d )2()1()()( +=

(17)

This is second-order in t; (exercise: prove it).

A wider class of schemes is furnished by so-calledpredictor-correctormethods which refine

their initialprediction with one (or more) corrections. A popular example of this type is theAdams-Bashforth-Moulton method:

predictor: ]5559379[ 1232411 nnnnnn

pred FFFFt +++=+

corrector: ]9195[ 1122411 ++ +++= npred

nnnnnFFFFt

Just as three-point advection schemes permit greater spatial accuracy than two-point schemes,

so the use of multiple time levels allows greater temporal accuracy. However, there are a

number of disadvantages which limit their application in CFD:

Storage; each computational variable has to be stored at all nodes at each time level. Start-up; initially, only data at time t= 0 is available; the first step inevitably requires

a single-step method.

6.4 Uses of Time-Marching in CFD

Time-dependent schemes are used in two ways:

(1) for a genuinely time-dependent problem;

(2) for time marching to steady state.

In case (1) accuracy and stability often impose restrictions on the time step and hence how

fast one can advance the solution in time. Because all nodal values must be updated at the

same rate the time step tis global; i.e. the same at all grid nodes.

In case (2) one is not seeking high accuracy so one simply adopts a stable algorithm, usually

Backward Differencing. Alternatively, if using an explicit scheme such as Forward

Differencing, the time step can be local, i.e. vary from cell to cell, in order to satisfy Courant-

number restrictions in each cell individually.

In practice, for incompressible flow, steady flow should be computable without time-

marching. This is not the case in compressible flow, where time-marching is necessary in

transonic calculations (flows with both subsonic and supersonic regions).

t

t(n+1)

(n+1)

(n)

(n-1) (n-2)

t(n)

t(n-1)

t(n-2)

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Summary The time-dependent fluid-flow equations are first-order in time and are solved by

time-marching.

Time-marching schemes may be explicit (time derivative known at the start of thetime step) or implicit(require iteration at each time step).

Common one-step methods are Forward Differencing (fully-explicit), BackwardDifferencing (fully-implicit) and Crank-Nicolson (semi-implicit).

One-step methods are easily implemented via changes to the matrix coefficients. Forthe Backward-Differencing scheme the only concessions required are:

t

Vbb

t

Vaa

old

P

PPPP

)(,

++

The only unconditionally-stable two-time-level scheme is Backward Differencing(fully-implicit timestepping). Other schemes have time-step restrictions: typically

limitations on the Courant number

x

tuc =

The Crank-Nicolson scheme is second-order accurate in t. The Backward-Differencing and Forward-Differencing schemes are both first order in t, which

means they need more time steps to achieve the same time accuracy.

Multi-step methods may be used to achieve accuracy and/or stability. However, theseare less favoured in CFD because of large storage overheads.

Time-accurate solutions require a global time step. A local time step may be used fortime-marching to steady state. In the latter case, high time accuracy is not required

and backward differencing is favoured as the most stable approach.

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Examples

Q1.

Use: (a) forward-differencing; (b) backward-differencing to solve the equation

0)0(,2d

d 2

==

tt numerically over the interval 0 t 1, using a timestep t= 0.25.

Q2.

Gears scheme for the approximation of a time derivative is

tt

nnnn

2

43

d

d )2()1()()( +=

where superscripts (n 2), (n 1), (n) denote successive time levels. Show that this scheme is

second-order accurate in time.

Q3.

The semi-discretised version of the scalar transport equation

sourcefluxoutwardnetamountt

=+)(d

d

over a control volume, centred at node P and containing fluid mass V, can be written

PFFPPP baaVt

=+ )(d

d

(a) Show that time integration using backward differencing, with a timestep t, can beimplemented by simple changes to the coefficients aP, {aF} and bP.

(b) Show that time integration using forward differencing is explicit, but imposes a

maximum timestep for stability, tmax.

(c) Show that time integration using the Crank-Nicolson method, with a timestep t, can

be implemented by changes to the coefficients aP, {aF} and bP and derive an

expression for the maximum stable timestep, tmax.

Q4. (Computational Hydraulics examination, May 2008 part)(a) Solve the first-order differential equation

1)0(,1d

d12

==

t

t

with a timestep t= 0.25 on the interval 0 t 1, using:

(i) forward differencing;

(ii) backward differencing.

(b) State (without mathematical detail) the advantages and disadvantages of using:

(i) forward-differencing

(ii) backward-differencingmethods in computational fluid dynamics.

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Selected Answers

Classroom Example 1

t

(a) Forwarddifferencing

(b) Backwarddifferencing

(c) Crank-Nicolson Exact( tet += 21 )

0 1 1 1 1

0.2 0.8 0.866667 0.836364 0.837462

0.4 0.68 0.788889 0.738843 0.740640

0.6 0.624 0.757407 0.695417 0.697623

0.8 0.6192 0.764506 0.696250 0.698658

1.0 0.65536 0.803755 0.733296 0.735759

Classroom Example 2

(a) Forward-differencing

t(old)

(old) t(new)

(new)0.00 1.0000 0.25 0.7500

0.25 0.7500 0.50 0.6484

0.50 0.6484 0.75 0.6115

0.75 0.6115 1.00 0.6598

(b) Backward-differencing:

t(old)

(old) t(new)

(new)0.00 1.0000 0.25 0.8502

0.25 0.8502 0.50 0.7681

0.50 0.7681 0.75 0.7627

0.75 0.7627 1.00 0.8559

(c) Crank-Nicolson:

t(old)

(old) t(new) (new)

0.00 1.0000 0.25 0.8104

0.25 0.8104 0.50 0.7156

0.50 0.7156 0.75 0.6960

0.75 0.6960 1.00 0.7737

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Q1.

(a) Forward differencing:

told

old tnew

new0.00 0 0.25 0

0.25 0 0.50 0.01563

0.50 0.01563 0.75 0.070310.75 0.07031 1.00 0.1758

(b) Backward differencing:

told

old tnew

new0.00 0 0.25 0.01042

0.25 0.01042 0.50 0.04861

0.50 0.04861 0.75 0.1262

0.75 0.1262 1.00 0.2508

Q4(a)

(i) Forward differencingoldt old newt new

0 1 0.25 1.2500

0.25 1.2500 0.50 1.5186

0.50 1.5186 0.75 1.6574

0.75 1.6574 1.00 1.4905

1.00 1.4905

(ii) Backward differencingoldt old newt new

0 1 0.25 1.2778

0.25 1.2778 0.50 1.4238

0.50 1.4238 0.75 1.3995

0.75 1.3995 1.00 1.2830

1.00 1.2830

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