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MA557/MA578/CS557 Lecture 28. Spring 2003 Prof. Tim Warburton 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 cs557 lecture 28

MA557/MA578/CS557Lecture 28

Spring 2003

Prof. Tim Warburton

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.








Necessary condition on qhat for stability


5 cont


Proof of possible solution

Proof of possible solution

Upwinding unveiled

Upwinding Unveiled

This is nothing more than upwinding in disguise

Equivalent conditions

Equivalent Conditions

Returning to dg scheme

Returning To DG Scheme



  • 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:

Tensor form

Tensor Form

Simplified tensor form

Simplified Tensor Form

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!!!

Example matlab code cont

Example Matlab Code cont

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.

The maxwell s equations we will use

The Maxwell’s Equations We Will Use

The maxwell s equations as conservation rule

The Maxwell’s Equations As Conservation Rule

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.



  • 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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