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 $u_t+a{u}_x=f(x)u+g(x)u^n$ and the wave equation $u_{tt}=f(x)u_{xx}$ that have a highly oscillating term like $f(x)=\sin (x/\epsilon ),\epsilon \ll 1$. 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.