Discontinuous Galerkin methods for hyperbolic and advection-dominated problems on surfaces

TL;DR

This paper extends the discontinuous Galerkin method to hyperbolic and advection-dominated problems on surfaces, providing stability analysis and optimal error estimates, verified through numerical experiments.

Contribution

It introduces a DG framework for surface problems, including a novel treatment of the discrete velocity field and stability analysis.

Findings

Optimal error estimates are proven under certain assumptions.

Numerical tests confirm the theoretical stability and accuracy.

The method effectively handles advection-dominated surface problems.

Abstract

We extend the discontinuous Galerkin (DG) framework to the analysis of first-order hyperbolic and advection-dominated problems posed on implicitly defined surfaces. The focus will be on the hyperbolic part, which is discretised using a "discrete surface" generalisation of the jump-stabilised upwind flux. A key issue arising in the analysis (which does not appear in the planar setting) is the treatment of the discrete velocity field, choices of which play an important role in the stability of the scheme. We then prove optimal error estimates in an appropriate norm given a number of assumptions on the discrete velocity field, which are then investigated and discussed in more detail. The theoretical results are verified numerically for a number of test problems exhibiting advection-dominated behaviour.