Eulerian methods for high frequency acoustics

The high frequency approximation to the wave equation yields an eikonal equation for the wave phase and a transport equation for the amplitude. Unfortunately, straightforward computation of the eikonal equation poses a challenge in that solutions may be multivalued which affects the convergence of a numerical scheme. Traditional numerical schemes invoke ray tracing methods to solve the eikonal equation in a Lagrangian frame of reference. Ray tracing performs well in many cases and handles the multivalued solution inherently. However, in long range simulations or simulations with complicated boundary conditions, ray solutions may perform poorly as rays are very sensitive to small perturbations in initial conditions. In the past few decades, several methods for solving the eikonal equation in a fixed (Eulerian) frame of reference have been devised. As with any computational method, each has its own advantages and disadvantages, but it is useful to be aware of alternatives since many problems in physical acoustics may benefit from having phase solutions computed on a fixed grid. We present a brief review of recent Eulerian approaches and discuss the merits and open issues with each.