A convergent finite difference scheme for the Ostrovsky-Hunter equation on a bounded domain

We prove the 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 periodic boundary conditions. The equation models, for example, shallow water waves in a rotating fluid and ultra-short light pulses in optical fibres, and its solutions can develop discontinuities in finite time. Our scheme extends monotone schemes for conservation laws to these equations. The convergence result also provides an existence proof for periodic entropy solutions of the general Ostrovsky-Hunter equation. Uniqueness and an $\mathcal{O}(\Dx^{1/2})$ bound on the $L^1$ error of the numerical solutions are shown using Kru\v{z}kov's technique of doubling of variables and a ``Kuznetsov type'' lemma. Numerical examples confirm the convergence results.