A central-upwind geometry-preserving method for hyperbolic conservation laws on the sphere

TL;DR

This paper presents a second-order, geometry-preserving central-upwind finite volume method for solving nonlinear hyperbolic conservation laws on the sphere, emphasizing high resolution, low dissipation, and simplicity without Riemann solvers.

Contribution

It introduces a novel semi-discrete scheme that is free of Riemann solvers, preserves geometric properties, and effectively captures discontinuous steady states on the sphere.

Findings

High accuracy in capturing discontinuous solutions

Low numerical dissipation observed in tests

Stable and efficient for complex steady states

Abstract

We introduce a second-order, central-upwind finite volume method for the discretization of nonlinear hyperbolic conservation laws posed on the two-dimensional sphere. The semi-discrete version of the proposed method is based on a technique of local propagation speeds and it is free of any Riemann solver. The main advantages of our scheme are the high resolution of discontinuous solutions, its low numerical dissipation, and its simplicity for the implementation. The proposed scheme does not use any splitting approach, which is applied in some cases to upwind schemes in order to simplify the resolution of Riemann problems. The semi-discrete form of the scheme is strongly linked to the analytical properties of the nonlinear conservation law and to the geometry of the sphere. The curved geometry is treated here in an analytical way so that the semi-discrete form of the proposed scheme is consistent with a geometric compatibility property. Furthermore, the time evolution is carried out by using a total-variation-diminishing Runge-Kutta method. A rich family of (discontinuous) stationary solutions is available for the problem under consideration when the flux is nonlinear and foliated (as identified by the author in an earlier work). We present here a series of numerical examples, obtained by considering non-trivial steady state solutions and this leads us to a good validation of the accuracy and efficiency of the proposed central-upwind finite volume method. Our numerical tests confirm the stability of the proposed scheme and clearly show its ability to capture accurately discontinuous steady state solutions to nonlinear hyperbolic conservation laws posed on the sphere.