Stanford 50: State of the Art and Future Directions of Computational Mathematics and Numerical Computing


  • March 29, 2007
  • 2:00 pm - 2:25 pm

High order one-step difference methods for wave propagation

Bertil Gustafsson (Stanford University)

We have earlier constructed high order explicit one-step difference methods for linear wave propagation problems with variable coefficients. They use staggered grids, and are norm conserving without any restriction on the coefficients other than boundedness. In particular, they can be used for wave propagation in discontinuous media, without any special treatment of the interior boundaries.

A special advantage is the effective implementation. Once the coefficients of the problem are defined at all grid points, the difference scheme is applied everywhere in the interior without modification. In recent work with B. Engquist, A-K. Tornberg and P. Wahlund, we have applied the same principle when treating real boundaries, like solid walls. The coefficients of the PDE system are given extreme values on one side of the boundary, and in this way the domain of interest can be embedded in a regular domain, keeping the effective implementation of the algorithm. The accuracy is formally brought down to first order because of the boundary treatment. This error is independent of time, and is in most cases dominated by the formally higher order phase error, which grows with time. However, we will show that one can modify the algorithm, for both interior and exterior boundaries, such that second order accuracy is obtained. This is done by a modification of the coefficients near the boundary, which means that the effective implementation is not destroyed.

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