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MA557/MA578/CS557 Lecture 28

MA557/MA578/CS557 Lecture 28. Spring 2003 Prof. Tim Warburton timwar@math.unm.edu. Linear Systems of 2D PDEs. A general linear pde system will look like:. Deriving DG. Ok – by popular demand here’s a brief derivation of the DG scheme for advection. 1. 2. 3. 4. 5.

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MA557/MA578/CS557 Lecture 28

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  1. MA557/MA578/CS557Lecture 28 Spring 2003 Prof. Tim Warburton timwar@math.unm.edu

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

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

  4. 3 4

  5. 5 Necessary condition on qhat for stability

  6. 5 cont 6

  7. Proof of possible solution

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

  9. Equivalent Conditions

  10. Returning To DG Scheme

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

  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. Tensor Form

  18. Simplified Tensor Form

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

  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 Sewe use umNtoF*(Fscale.*fC) where: Fscale = sjac./(umNtoF*jac);

  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. Example Matlab Code cont

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

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

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

  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. The Maxwell’s Equations We Will Use

  28. The Maxwell’s Equations As Conservation Rule

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

  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.

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