A convergent finite difference method for a nonlinear variational wave equation

TL;DR

This paper proves the convergence of a semi-discrete upwind finite difference scheme for a nonlinear variational wave equation, transforming it into Riemann invariants and analyzing the scheme's stability under specific conditions.

Contribution

The paper introduces a convergent semi-discrete upwind scheme for the nonlinear variational wave equation using Riemann invariants, with rigorous proof under certain assumptions.

Findings

The scheme converges for positive, increasing wave speed functions.

Numerical examples demonstrate the scheme's effectiveness.

The method handles nonlinearity and variational structure effectively.

Abstract

We establish rigorously convergence of a semi-discrete upwind scheme for the nonlinear variational wave equation $u_{tt} - c(u)(c(u) u_x)_x = 0$ with $u|_{t=0}=u_0$ and $u_t|_{t=0}=v_0$. Introducing Riemann invariants $R=u_t+c u_x$ and $S=u_t-c u_x$, the variational wave equation is equivalent to $R_t-c R_x=\tilde c (R^2-S^2)$ and $S_t+c S_x=-\tilde c (R^2-S^2)$ with $\tilde c=c'/(4c)$. An upwind scheme is defined for this system. We assume that the the speed $c$ is positive, increasing and both $c$ and its derivative are bounded away from zero and that $R|_{t=0}, S|_{t=0}\in L^1\cap L^3$ are nonpositive. The numerical scheme is illustrated on several examples.