Spring 2003

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MA557/MA578/CS557Lecture 28

• Spring 2003
• Prof. Tim Warburton
• timwar_at_math.unm.edu

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Linear Systems of 2D PDEs

• A general linear pde system will look like

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Deriving DG

• Ok by popular demand heres a brief derivation
  of the DG scheme for advection.

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2
4
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4
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Necessary condition on qhat for stability
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5 cont
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Proof of possible solution
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Upwinding Unveiled
This is nothing more than upwinding in disguise
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Equivalent Conditions
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Returning To DG Scheme
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Convergence

• Convergence follows as beforehowever 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.

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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

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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

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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

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Lax-Friedrichs DG Scheme For Linear Systems
As before we can generalize the scalar PDE
scheme to a system
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Discrete LF/DG Scheme

• We choose to discretize the Pp polynomial space
  on each triangle using M(p1)(p2)/2 Lagrange
  interpolatory polynomials.
• i.e.
• The scheme now reads

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Tensor Form
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Simplified Tensor Form
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Geometric Factors

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

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Surface Terms

• In the Matlab code first we extract the nodes on
  the edges of the elements
• fC umFtoNC
• The resulting matrix fC has dimension
  (umNfaces(umP1))xumNel
• This lists all nodes from element 1, edge 1 then
  element 1, edge 2 then
• In order to multiply, say fC, by Se we use
  umNtoF(Fscale.fC) where Fscale
  sjac./(umNtoFjac)

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Example Matlab Code(Upwind DG Advection Scalar
Eqn)

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

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Example Matlab Code cont
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Specific Example

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

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Maxwells Equations (TM mode)

• In the absence of sources, Maxwells equations
  are
• Where

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Free Space..

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

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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.

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The Maxwells Equations We Will Use
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The Maxwells Equations As Conservation Rule
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TM Maxwells Boundary Condition

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

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The Maxwells 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.

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Eigenvalues

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

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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 Maxwells
• 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.