Long range numerical simulation of short waves as nonlinear solitary waves

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 functions according to the wave equation. These basis functions are generated as stable solitary waves where 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 thin nonlinear solitary waves and remain 2–3 grid cells in thickness indefinitely 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), defined on a regular grid, rather than collections of markers. In this way, the complex logic of current ray tracing methods, which involves allocation of markers to each surface and interpolation as the markers separate, is avoided.