MA557/MA578/CS557 Lecture 28

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MA557/MA578/CS557 Lecture 28. Spring 2003 Prof. Tim Warburton [email protected] 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/CS557Lecture 28

Spring 2003

Prof. Tim Warburton

[email protected]

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

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.