1. Numerical Methods for Partial Differential Equations CAAM 452 Spring 2005 Lecture 12 Instructor: Tim Warburton

2. CAAM 452 Spring 2005 Godunov Scheme Summary To complete this scheme we now specify how to compute the slopes. Standard formulae:

3. CAAM 452 Spring 2005 With Limiting Minmod slope limiter: Monotonized central-difference limiter (MC limiter)

4. CAAM 452 Spring 2005 Today More limiters Flux limiting function formulation. We will discuss Harten’s sufficient conditions for a numerical method (including limiter) to be TVD Sweby TVD diagrams for flux limiting functions. Extension to systems of linear PDE’s Extension to nonlinear PDE’s

5. CAAM 452 Spring 2005 Flux Formulation with Piecewise Linear Reconstruction Last time we showed how the ansatz of a piecewise linear reconstruction and Godunov’s method allowed us to compute the time averaged flux contribution at each end of the I’th cell Notice: we can obtain the i-1/2 flux by setting i->i-1 in the i+1/2 flux formula (i.e. the flux formula is continuous at the cell boundary)

6. CAAM 452 Spring 2005 cont Using this notation the scheme becomes: This is known as the flux formulation with piecewise reconstruction.

7. CAAM 452 Spring 2005 cont So far we have assumed u>0 but we can generalize this for u<0 using the same approach as before: To simplify this we write it as:

8. CAAM 452 Spring 2005 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.

9. CAAM 452 Spring 2005 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:

10. CAAM 452 Spring 2005 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.

11. CAAM 452 Spring 2005 cont Using this formulation we can recover the methods we have seen before and some new limiters:

12. CAAM 452 Spring 2005 cont Using this notation we can write down the scheme in terms of the flux limiter function ( ): Upwind scheme flux contibution Limited downwind cell interface flux contribution Limited upwind cell interface flux contribution u>0 u<0

13. CAAM 452 Spring 2005 Harten’s 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:

14. CAAM 452 Spring 2005 Sweby Diagrams http://locus.siam.org/fulltext/SINUM/volume-21/0721062.pdf http://locus.siam.org/fulltext/SINUM/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:

15. CAAM 452 Spring 2005 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

16. CAAM 452 Spring 2005 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. Lax-Wendroff FrommBeam-Warming 123 1 2

17. CAAM 452 Spring 2005 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: 123 1 2 http://locus.siam.org/fulltext/SINUM/volume-21/0721062.pdf

18. CAAM 452 Spring 2005 Minmod Flux Limiter on Sweby Diagram 123 1 2 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).

19. CAAM 452 Spring 2005 Superbee Flux Limiter on Sweby Diagram 123 1 2 The Superbee limiter applies the minimum limiting and maximum steepening possible to remain TVD. It is known to suffer from excessive sharpening of slopes 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.

20. CAAM 452 Spring 2005 MC Flux Limiter on Sweby Diagram 123 1 2 The MC limiter transitions from upwind (theta<0) to Fromm (at theta=1/3) then switches to a constant(at theta=3). This is a compromise between Superbee and minmod

21. CAAM 452 Spring 2005 van Leer Flux Limiter The van Leer limiter charts a careful compromise path through the Sweby TVD region.

22. CAAM 452 Spring 2005 Summary of Some Flux Limiting Functions Nonlinear second order TVD limiters Linear non-TVD limiters

23. CAAM 452 Spring 2005 Implementation For u>0 the scheme looks like: We can easily achieve this in matlab:

24. CAAM 452 Spring 2005 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.edu/~caam452/CodeSnip pets/fluxlimiter.m With the initial condition supplied by: http://www.caam.rice.edu/~caam452/CodeSnip pets/fluxlimiterexact.m

25. CAAM 452 Spring 2005 Homework 4 Q1) Using N=80,160,320,640,1280 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(pi*x) ii. sin(pi*x) + (abs(x-.5)<.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.

26. CAAM 452 Spring 2005 Homework 4 cont Q2a) Invent your own 2 nd 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.