Stability and convergence of a conservative finite difference scheme for the modified Hunter--Saxton equation

TL;DR

This paper develops and rigorously analyzes a conservative finite difference scheme for the modified Hunter--Saxton equation, ensuring stability and convergence despite the challenges posed by its mixed derivative and complex behavior.

Contribution

It introduces a new conservative finite difference scheme for the modified Hunter--Saxton equation and proves its stability and uniform convergence.

Findings

The scheme is stable in the uniform norm.

The scheme converges uniformly to smooth solutions.

Discrete conservation laws are effectively used to handle mixed derivatives.

Abstract

The modified Hunter--Saxton equation models the propagation of short capillary-gravity waves. As it involves a mixed derivative, its initial value problem on the periodic domain is much more complicated than the standard evolutionary equations. Although its local well-posedness has recently been proved, the behavior of its solution is yet to be investigated. In this paper, to develop a reliable numerical method for this problem, we derive a conservative finite difference scheme. Then, we rigorously prove not only its stability in the sense of the uniform norm but also its uniform convergence to sufficiently smooth exact solutions. Discrete conservation laws are used to overcome the difficulty due to the mixed derivative.