1. 1 Spring 2003 Prof. Tim Warburton timwar@math.unm.edu MA557/MA578/CS557 Lecture 28

2. 2 Linear Systems of 2D PDEs A general linear pde system will look like:

3. 3 Deriving DG Ok – by popular demand here’s a brief derivation of the DG scheme for advection. 1 2

4. 4 3 4

5. 5 5 Necessary condition on qhat for stability

6. 6 5 cont 6

7. 7 Proof of possible solution

8. 8 Upwinding Unveiled This is nothing more than upwinding in disguise

9. 9 Equivalent Conditions

10. 10 Returning To DG Scheme

11. 11 Convergence Convergence follows as before…however for a given choice of qhat we need to verify consistency to guarantee stability. In fact for the convergence result to hold we required the truncation error to vanish in the limit of small h and also large p. For example we required: This is easily verified for the choice of qhat we just made.

12. 12 General Choice of qhat We established the following sufficient conditions on the choice of the function qhat(q+,q-): As long as these are met for a choice of qhat then the scheme will be stable. However, the choice of qhat may impact accuracy…

13. 13 An Alternative Construction for qhat Exercise – prove that this is a sufficient condition for stability. i.e. This generates the Lax-Friedrich flux based DG scheme:

14. 14 Lax-Friedrichs Fluxes Suppose we know the maximum wave speed of the hyperbolic system: As long as the matrix are codiagonalizable for any linear combination. i.e. Has real eigenvalues for any real alpha,beta. And: then we can use the following DG scheme:

15. 15 Lax-Friedrichs DG Scheme For Linear Systems As before – we can generalize the scalar PDE scheme to a system:

16. 16 Discrete LF/DG Scheme We choose to discretize the Pp polynomial space on each triangle using M=(p+1)(p+2)/2 Lagrange interpolatory polynomials. i.e. The scheme now reads:

17. 17 Tensor Form

18. 18 Simplified Tensor Form

19. 19 Geometric Factors We use the chain rule to compute the Dx and Dy matrices: In the umSCALAR2d scripts the

20. 20 Surface Terms In the Matlab code first we extract the nodes on the edges of the elements: fC = umFtoN*C The resulting matrix fC has dimension (umNfaces*(umP+1))xumNel This lists all nodes from element 1, edge 1 then element 1, edge 2 then… In order to multiply, say fC, by S e we use umNtoF*(Fscale.*fC) where: Fscale = sjac./(umNtoF*jac);

21. 21 Example Matlab Code (Upwind DG Advection Scalar Eqn) This is not strictly a global Lax-Friedrichs scheme since the lambda is chosen locally!!!

22. 22 Example Matlab Code cont

23. 23 Specific Example We will try out this scheme on the 2D, transverse mode, Maxwell’s equations.

24. 24 Maxwell’s Equations (TM mode) In the absence of sources, Maxwell’s equations are: Where

25. 25 Free Space.. For simplicity we will assume that the permeability and permittivity are =1 we then obtain:

26. 26 Divergence Condition We next notice that by taking the x derivative of the first equation and the y derivative of the second equation we find that: i.e. the fourth equation (divergence condition) is a natural result of the equations – assuming that the initial condition for the magnetic field H is divergence free.

27. 27 The Maxwell’s Equations We Will Use

28. 28 The Maxwell’s Equations As Conservation Rule

29. 29 TM Maxwell’s Boundary Condition We will use the following PEC boundary condition – which corresponds to the boundary being a perfectly, electrical conducting material:

30. 30 The Maxwell’s Equations In Matrix Form We are now assuming that the magnetic field is going to be divergence free – so we just neglect this for the moment.

31. 31 Eigenvalues Then: C has eigenvalues which satisfy: So the matrices are co-diagonalizable for all real alpha, beta and

32. 32 Project Time Due 04/11/03 Code up a 2D DG solver for an arbitrary hyperbolic system based on the 2D DG advection code provided. Test it on TM Maxwell’s –Set A and B as described, set lambdatilde = 1 and test on some domain of your choosing. –Use PEC boundary conditions all round. Test it on a second set of named equations of your own choosing (determine these equations by research) Compute convergence rates.. For smooth solutions.