Capturing short waves over long distances as nonlinear solitary waves

We present a new numerical method, “Wave Confinement” (WC), to efficiently solve the linear wave equation. The novelty of our approach is in the use of nonlinear solitary waves to accommodate long range wave propagation of short pulses with no numerical diffusion or dispersion. The method involves extension of the wave equation by adding a nonlinear term, which does not interfere with conservation of the important integral quantities such as total amplitude or propagation of the pulse centroid. The main idea is to create an “extended” partial differential equation where the basic entities are stable co-dimension one surfaces. The computed pulses remain indefinitely concentrated over only 2-4 grid cells (in thickness). The main focus of this paper is to show the accuracy of WC to capture caustic regions and the simplicity to include scattering from complex surfaces.