Solution Interpolation Method for Highly Oscillating Hyperbolic Equations

Abstract

This paper deals with a novel numerical scheme for hyperbolic equations with rapidly changing terms. We are especially interested in the quasilinear equation u(t) + au(x). = f(x)u + g(x)u(n) and the wave equation u(tt) = f(x)u(xx) that have a highly oscillating term like f(x) = sin(x/epsilon), epsilon << 1. It also applies to the equations involving rapidly changing or even discontinuous coefficients. The method is based on the solution interpolation and the underlying idea is to establish a numerical scheme by interpolating numerical data with a parameterized solution of the equation. While the constructed numerical schemes retain the same stability condition, they carry both quantitatively and qualitatively better performances than the standard method.