Hyperbolic conservation laws on the sphere. A geometry-compatible finite volume scheme

TL;DR

This paper develops a geometry-compatible finite volume scheme for scalar hyperbolic conservation laws on the sphere, enabling accurate numerical solutions and analysis of various flux vector classes, including periodic and stationary solutions.

Contribution

It introduces a novel finite volume scheme on a web-like mesh that respects geometric compatibility for conservation laws on the sphere, extending previous well-posedness results.

Findings

Constructed equatorial periodic solutions on the sphere.

Developed a scheme that preserves steady states and supports confined solutions.

Presented numerical examples demonstrating the scheme's effectiveness.

Abstract

We consider entropy solutions to the initial value problem associated with scalar nonlinear hyperbolic conservation laws posed on the two-dimensional sphere. We propose a finite volume scheme which relies on a web-like mesh made of segments of longitude and latitude lines. The structure of the mesh allows for a discrete version of a natural geometric compatibility condition, which arose earlier in the well-posedness theory established by Ben-Artzi and LeFloch. We study here several classes of flux vectors which define the conservation law under consideration. They are based on prescribing a suitable vector field in the Euclidean three-dimensional space and then suitably projecting it on the sphere's tangent plane; even when the flux vector in the ambient space is constant, the corresponding flux vector is a non-trivial vector field on the sphere. In particular, we construct here "equatorial periodic solutions", analogous to one-dimensional periodic solutions to one-dimensional conservation laws, as well as a wide variety of stationary (steady state) solutions. We also construct "confined solutions", which are time-dependent solutions supported in an arbitrarily specified subdomain of the sphere. Finally, representative numerical examples and test-cases are presented.