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MA557/MA578/CS557 Lecture 28

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- A general linear pde system will look like:

- Ok – by popular demand here’s a brief derivation of the DG scheme for advection.

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Necessary condition on qhat for stability

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This is nothing more than upwinding in disguise

- 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.

- 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…

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

- 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:

As before – we can generalize the scalar PDE scheme to a system:

- 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:

- We use the chain rule to compute the Dx and Dy matrices:
- In the umSCALAR2d scripts the

- 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);

- This is not strictly a global Lax-Friedrichs scheme since the lambda is chosen locally!!!

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

- In the absence of sources, Maxwell’s equations are:
- Where

- For simplicity we will assume that the permeability and permittivity are =1 we then obtain:

- 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.

- We will use the following PEC boundary condition – which corresponds to the boundary being a perfectly, electrical conducting material:

- We are now assuming that the magnetic field is going to be divergence free – so we just neglect this for the moment.

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

- 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.