A convergent finite difference scheme for the Ostrovsky--Hunter equation with Dirichlet boundary conditions

TL;DR

This paper proves the convergence of a finite difference scheme for the Ostrovsky--Hunter equation with Dirichlet boundary conditions, establishing existence, uniqueness, and providing numerical validation.

Contribution

It extends monotone schemes for conservation laws to the Ostrovsky--Hunter equation and proves convergence, existence, and uniqueness of entropy solutions.

Findings

Convergence of the scheme to the entropy solution

Existence of entropy solutions for the initial-boundary value problem

Numerical experiments confirming convergence and rates

Abstract

We prove convergence of a finite difference scheme to the unique entropy solution of a general form of the Ostrovsky--Hunter equation on a bounded domain with non-homogeneous Dirichlet boundary conditions. Our scheme is an extension of monotone schemes for conservation laws to the equation at hand. The convergence result at the center of this article also proves existence of entropy solutions for the initial-boundary value prob lem for the general Ostrovsky--Hunter equation. Additionally, we show uniqueness using Kru\v{z}kov's doubling of variables technique. We also include numerical examples to confirm the convergence results and determine rates of convergence experimentally.