Heuristic Modified Equation Analysis on Oscillations in Numerical Solutions of Conservation Laws

Abstract

Oscillations are ubiquitous in numerical solutions obtained by high order or even first order schemes for hyperbolic problems and are conventionally understood as the consequence of low dissipation effects of underlying numerical schemes. Earlier analysis was done mainly through the effective discrete Fourier analysis for linear problems or the modified equation approach in smooth solution regions. In this paper, a so-called heuristic modified equation is derived when applied to nonlinear problems, particularly for oscillatory modes of solutions whose counterpart in linear problems are high frequency mode solutions, and the dissipation effect is distinguished as a numerical damping and a numerical diffusion. The former is reflected through the zero order term of the heuristic modified equation and the latter through the second order differential term. It turns out that the effect of dissipation is categorized as a damping, a neutrality, and an amplification, and that the numerical damping plays a dominant role in offsetting the oscillatory modes. When the amplification effect is taken, the numerical scheme often comes unstable.