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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

MA557/MA578/CS557Lecture 28

Spring 2003

Prof. Tim Warburton

[email protected]

linear systems of 2d pdes
Linear Systems of 2D PDEs
  • A general linear pde system will look like:
deriving dg
Deriving DG
  • Ok – by popular demand here’s a brief derivation of the DG scheme for advection.

1

2

slide4

3

4

slide5

5

Necessary condition on qhat for stability

upwinding unveiled
Upwinding Unveiled

This is nothing more than upwinding in disguise

convergence
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
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
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
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
Lax-Friedrichs DG Scheme For Linear Systems

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

discrete lf dg scheme
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:
geometric factors
Geometric Factors
  • We use the chain rule to compute the Dx and Dy matrices:
  • In the umSCALAR2d scripts the
surface terms
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
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
Specific Example
  • We will try out this scheme on the 2D, transverse mode, Maxwell’s equations.
maxwell s equations tm mode
Maxwell’s Equations (TM mode)
  • In the absence of sources, Maxwell’s equations are:
  • Where
free space
Free Space..
  • For simplicity we will assume that the permeability and permittivity are =1 we then obtain:
divergence condition
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
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
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
Eigenvalues
  • Then:
  • C has eigenvalues which satisfy:
  • So the matrices are co-diagonalizable for all real alpha, beta and
project time
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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