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


  • March 30, 2007
  • 9:25 am - 9:50 am

Computational wave propagation in bounded and unbounded domains

Marcus Grote (Universität Basel)

The accurate and reliable simulation of wave phenomena is of fundamental importance in a wide range of engineering applications such as fiber optics, wireless communication, sonar and radar technology, non-invasive testing, ultra-sound imaging, and optical microscopy. To address the wide range of difficulties involved, we consider symmetric interior penalty discontinuous Galerkin (IP-DG) methods, which easily handle elements of various types and shapes, irregular non-matching grids, and even locally varying polynomial order. Moreover, in contrast to standard (conforming) finite element methods, IP-DG methods yield an essentially diagonal mass matrix; hence, when coupled with explicit time integration, the overall numerical scheme remains truly explicit in time. To circumvent the stability (CFL) condition imposed on the time step by the smallest elements in the underlying mesh, we further propose energy conserving explicit local time-stepping schemes.

For problems set in an unbounded domain, an artificial boundary is required to confine the region of interest to a finite computational domain. Then, a nonreflecting boundary condition is required at the artificial boundary, which avoids spurious reflections from it. When a scatterer consists of several components, the use of a single artificial boundary to enclose the entire region of interest becomes too expensive. Instead, it is preferable to embed each component of the scatterer in a separate sub-domain. As waves may bounce back and forth between domains, they are no longer purely outgoing outside the computational domain, so that most standard approaches cannot be used. To overcome this difficulty, we show how to devise exact nonreflecting boundary conditions for multiple scattering problems, which avoid spurious reflections from the artificial boundary.

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