# MA557/MA578/CS557 Lecture 28 - PowerPoint PPT Presentation

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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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## MA557/MA578/CS557Lecture 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

Necessary condition on qhat for stability

5 cont

6

### Upwinding Unveiled

This is nothing more than upwinding in disguise

### 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.

### 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…

### 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:

### 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:

### Lax-Friedrichs DG Scheme For Linear Systems

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

### 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.

### Geometric Factors

• We use the chain rule to compute the Dx and Dy matrices:

• In the umSCALAR2d scripts the

### 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);

### Example Matlab Code(Upwind DG Advection Scalar Eqn)

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

### Specific Example

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

### Maxwell’s Equations (TM mode)

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

• Where

### Free Space..

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

### 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.

### TM Maxwell’s Boundary Condition

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

### 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.

### Eigenvalues

• Then:

• C has eigenvalues which satisfy:

• So the matrices are co-diagonalizable for all real alpha, beta and

### 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.