Discontinuous Galerkin methods for hyperbolic and advection-dominated problems on surfaces
This work provides a theoretical foundation for solving hyperbolic PDEs on surfaces, which is important for applications in fluid dynamics and material science, but the analysis is incremental as it builds on existing DG methods.
The authors extend the discontinuous Galerkin framework to hyperbolic and advection-dominated problems on implicitly defined surfaces, proving optimal error estimates under assumptions on the discrete velocity field, and verifying results numerically.
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.