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Numerical Methods for Partial Differential

Equations

- CAAM 452
- Spring 2005
- Lecture 12
- Instructor Tim Warburton

Godunov Scheme Summary

- To complete this scheme we now specify how to

compute the slopes. - Standard formulae

With Limiting

Minmod slope limiterMonotonized

central-difference limiter (MC limiter)

Today

- More limiters
- Flux limiting function formulation.
- We will discuss Hartens sufficient conditions

for a numerical method (including limiter) to be

TVD - Sweby TVD diagrams for flux limiting functions.
- Extension to systems of linear PDEs
- Extension to nonlinear PDEs

Flux Formulation with Piecewise Linear

Reconstruction

- Last time we showed how the ansatz of a piecewise

linear reconstruction and Godunovs method

allowed us to compute the time averaged flux

contribution at each end of the Ith cell

Notice we can obtain the i-1/2 flux by setting

i-gti-1 in the i1/2 flux formula (i.e. the flux

formula is continuous at the cell boundary)

cont

- Using this notation the scheme becomes
- This is known as the flux formulation with

piecewise reconstruction.

cont

- So far we have assumed ugt0 but we can generalize

this for ult0 using the same approach as before - To simplify this we write it as

cont

- By writing the time interval averaged flux

function in this way - We are philosophically moving away from a local

cell reconstruction approach towards controlling

the flux contribution from jumps in the averages

between elements.

Flux Limiters

- The idea is limit the flux of q between cells

and you will subsequently limit spurious growth

in the cell averages near discontinuities - A general approach is to multiply the jump in

cell averages by a limiting function

cont

- The theta ratio can be thought of as a smoothness

indicator near the cell interface at x_i-1/2. - If the data is smooth we expect the ratio to be

approximately 1 (except at extrema) - Near a discontinuity we expect the ratio to be

far away from 1. - The flux limiting function, phi, will range

between 0 and 2. The smaller it is, the more

limiting is applied to a jump in cell averages.

Above 1 it is being used to steepen the effective

reconstruction.

cont

- Using this formulation we can recover the methods

we have seen before and some new limiters

cont

- Using this notation we can write down the scheme

in terms of the flux limiter function (

)

ugt0

Limited downwindcell interface fluxcontribution

Limited upwind cell interfaceflux contribution

Upwind schemeflux contibution

ult0

Hartens Theorem

- Theorem Consider a general method of the form
- for one time step, where the coefficients C and

D are arbitrary values (which in particular may

depend on qbar in some way). - Then provided

that the following conditions are satisfied

Sweby Diagramshttp//locus.siam.org/fulltext/SINU

M/volume-21/0721062.pdf

- We need to express the flux limited scheme
- In the form
- And then satisfying the Harten conditions will

guarantee the method is TVD. - An appropriate choice (which we can work with) is

cont

- In this case since the D coefficients are zero

and the Harten TVD conditions reduce to - This will hold if
- We can summarize this in terms of the minmod

function - In addition we require
- See LeVeque p 116-118 for details

cont

i.e. any flux limiting function must

satisfy to be TVD. Graphically, the shaded

region is the TVD region Clearly non

of these linear limiters generate a TVD scheme.

Fromm

Beam-Warming

2

1

Lax-Wendroff

1

2

3

cont

- To guarantee second order accuracy and avoid

excessive compression of solutions, Sweby

suggested the following reduced portion of the

TVD region as a suitable range for the flux

limiting function

2

1

1

2

3

http//locus.siam.org/fulltext/SINUM/volume-21/072

1062.pdf

Minmod Flux Limiter on Sweby Diagram

2

1

1

2

3

It is apparent that the minmod flux limiter

applies the maximum possible limiting allowed

within the second order TVD region. (i.e. it will

be rather dissipative and smear out

discontinuities somewhat as seen on the right

hand side figure).

Superbee Flux Limiter on Sweby Diagram

2

1

1

2

3

The Superbee limiter applies the minimum limiting

and maximum steepeningpossible to remain TVD. It

is known to suffer from excessive sharpening

ofslopes as a result. On the right we show what

happens to a smooth sine wave after 20

periods. Notice the flattening of the peaks and

the steepening of the slopes.

MC Flux Limiter on Sweby Diagram

2

1

1

2

3

The MC limiter transitions from upwind (thetalt0)

to Fromm (at theta1/3) then switches to a

constant(at theta3). This is a compromise

between Superbee and minmod

van Leer Flux Limiter

The van Leer limiter charts a careful compromise

path throughthe Sweby TVD region.

Summary of Some Flux Limiting Functions

Linear non-TVD limiters

Nonlinear second orderTVD limiters

Implementation

- For ugt0 the scheme looks like
- We can easily achieve this in matlab

Matlab Version

This is a sample Matlab implementation of a

piecewise linear reconstructed Godunov approach

with a selection of flux limiters. Available

from the course home page http//www.caam.rice.e

du/caam452/CodeSnippets/fluxlimiter.m With the

initial condition supplied by http//www.caam.ri

ce.edu/caam452/CodeSnippets/fluxlimiterexact.m

Homework 4

- Q1) Using N80,160,320,640,1280 estimate the

solution order of accuracy of the - flux limited scheme
- with flux limiting functions
- i. Fromm
- ii. minmod
- iii. MC
- using initial conditions
- i. sin(pix)
- ii. sin(pix) (abs(x-.5)lt.25)
- on the periodic interval -1,1).
- Use the fluxlimiter.m Matlab code from the web

page. - You will also need to download fluxlimiterexact.m

and minmod.m - Measure error both using the maxmimum norm, l2

norm and finally the - maximum norm with data points near the

discontinuity excluded. - Use error plots and tables with discussion to

describe your results.

Homework 4 cont

- Q2a) Invent your own 2nd order TVD flux limiter

function (i.e. a function with range contained in

the Sweby TVD region) - Q2b) Modify sweby.m to plot your flux limiter

function and compare with the limiter functions

already used. - Q2c) Estimate order of accuracy for a smooth

initial condition to the advection equation - Q2d) Estimate order of accuracy for a

discontinuous initial condition to the advection

equation - Q2e) Compare results (with diagrams,results and

comments) and discuss how your limiter differs

from the other limiters we have seen.