A New Eulerian Method for the Computation of Propagating Short Acoustic and Electromagnetic Pulses

A new method is described to compute short acoustic or electromagnetic pulses that propagate according to geometrical optics. The pulses are treated as zero thickness sheets that can propagate over long distances through inhomogeneous media with multiple reflections. The method has many of the advantages of Lagrangian ray tracing, but is completely Eulerian, typically using a uniform Cartesian grid. Accordingly, it can treat arbitrary configurations of pulses that can reflect from surfaces and pass through each other without requiring special computational marker arrays for each pulse. Also, information describing the pulses, which are treated as continuous surfaces, can be available throughout the computational grid, rather than only at isolated individual markers. The method uses a new type of representation, which we call “Dynamic Surface Extension.” The basic idea is to propagate or “broadcast” defining fields from each pulse surface through a computational grid. These fields carry information about a nearby pulse surface that is used at each node to compute the location of the pulse surfaces and other attributes, such as amplitude. Thus the emphasis is on the dynamics of these propagating defining fields, which obey only local Eulerian equations at each node. The Dynamic Surface Extension representation can be thought of as dual to level set representation: The defining fields involve single valued variables which are constant at each time along lines that are normal to the evolving surface, whereas level set techniques involve a function which has constant values on the evolving surface and neighboring surfaces. In this way the new method overcomes the inability of level set or Eikonal methods to treat intersecting pulses that obey a wave equation and can pass through each other, while still using only single-valued variables. Propagating thin pulse surfaces in 1-D, 2-D, and 3-D that can reflect from boundaries and pass through each other are computed using the new method. The method was first presented as a new, general representation of surfaces, filaments, and particles by J. Steinhoff and M. Fan (1998, Eulerian computation of evolving surfaces, curves and discontinuous fields, UTSI preprint).