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/CS557 Lecture 28

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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


5 cont

6


Proof of possible solution


Upwinding Unveiled

This is nothing more than upwinding in disguise


Equivalent Conditions


Returning To DG Scheme


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.

  • The scheme now reads:


Tensor Form


Simplified Tensor Form


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!!!


Example Matlab Code cont


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.


The Maxwell’s Equations We Will Use


The Maxwell’s Equations As Conservation Rule


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.


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