Convergence rates of finite difference schemes for the linear advection and wave equation with rough coefficient

TL;DR

This paper establishes convergence rates for explicit finite difference schemes solving the linear advection and wave equations with rough coefficients, linking rates to coefficient regularity and initial data continuity, supported by numerical experiments.

Contribution

It provides the first explicit convergence rate analysis for finite difference schemes with H"older continuous coefficients in one dimension.

Findings

Convergence rates depend on H"older regularity of coefficients.

Numerical experiments confirm theoretical convergence rates.

Rates vary with initial data modulus of continuity.

Abstract

We prove convergence rates of explicit finite difference schemes for the linear advection and wave equation in one space dimension with H\"older continuous coefficient. The obtained convergence rates explicitly depend on the H\"older regularity of the coefficient and the modulus of continuity of the initial data. We compare the theoretically established rates with the experimental rates of a couple of numerical examples.