A non-negativity-preserving cut-cell discontinuous Galerkin method for the diffusive wave equation

TL;DR

This paper introduces a non-negativity-preserving cut-cell discontinuous Galerkin method for the diffusive wave equation, capable of handling complex bathymetry and meshes, demonstrating higher accuracy than finite volume methods.

Contribution

The paper presents a novel discontinuous Galerkin method that preserves non-negativity and achieves second-order accuracy for the diffusive wave equation, outperforming finite volume methods.

Findings

DG method is fully second-order accurate on analytical solutions.

Finite volume method is only first-order accurate.

DG method requires fewer mesh refinements to match solutions.

Abstract

A non-negativity-preserving cut-cell discontinuous Galerkin method for the degenerate parabolic diffusive wave approximation of the shallow water equation is presented. The method can handle continuous and discontinuous bathymmetry as well as general triangular meshes. It is complemented by a finite volume method on Delauney triangulations which is also shown to be non-negativity preserving. Both methods feature an upwind flux and can handle Manning's and Chezy's friction law. By numerical experiment we demonstrate the discontinuous Galerkin method to be fully second-order accurate for the Barenblatt analytical solution on an inclined plane. In constrast, the finite volume method is only first-order accurate. Further numerical experiments show that three to four mesh refinements are needed for the finite volume method to match the solution of the discontinuous Galerkin method.