Nonlinear Localized Dissipative Structures for Solving Wave Equations over Long Distances

Abstract

A new numerical method has been developed to propagate short wave equation pulses over indefinite distances and through regions of varying index of refraction, including multiple reflections. The method, “Wave Confinement”, utilizes a newly developed nonlinear partial differential equation that propagates basis function according to the wave equation. The discretized equation can be solved without any numerical dissipation. The method can also be used to solve for harmonic waves in the high frequency (Eikonal) limit, including multiple arrivals. The solution involves discretizing the wave equation on a uniform Eulerian grid and adding a simple nonlinear “Confinement” term. This term does not change the amplitude (integrated through each point on the pulse surface) or the propagation velocity, or arrival time, and yet results in capturing the waves as thin surfaces that propagate as nonlinear solitary waves and remain 2–3 grid cells in thickness with no numerical spreading. With the method, only a simple discretized equation is solved each time step at each grid node. The method can be contrasted to Lagrangian Ray Tracing: it is an Eulerian based method that captures the waves directly on the computational grid, where the basic objects are codimension 1 surfaces (in the fine grid limit), rather than rays. In this way, the complex logic of current ray tracing methods, which involve allocation of markers to each surface and interpolation as the markers separate, is avoided. With the new method, even though the surfaces can pass through each other and involve a nonlinear term, there is no interaction effect from this term on the variables of interest, allowing the linear wave equation to be accurately simulated.