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

Panasun Manorost, Peter Bastian

TL;DR

The paper introduces a non-negativity-preserving cut-cell discontinuous Galerkin method (CUDGM) for the diffusive wave equation (DWE), capable of handling continuous or discontinuous bathymetry on unstructured triangular meshes and wet/dry fronts. It compares CUDGM to a Voronoi finite-volume method (VFVM), both with upwind fluxes, demonstrating that CUDGM achieves second-order accuracy on challenging front dynamics while preserving nonnegativity, even on coarse meshes. A cut-cell formulation avoids the small-cell problem and enables robust front-tracking, with regularization ensuring stability near dry states. Numerical tests on Barenblatt, obstacle, and dambreak scenarios show that CUDGM can match or exceed VFVM accuracy at lower computational cost on coarse meshes and remains effective for realistic bathymetric data. The approach provides a practical, high-accuracy tool for flood modelling with complex terrain and wet/dry interfaces.

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.

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

TL;DR

The paper introduces a non-negativity-preserving cut-cell discontinuous Galerkin method (CUDGM) for the diffusive wave equation (DWE), capable of handling continuous or discontinuous bathymetry on unstructured triangular meshes and wet/dry fronts. It compares CUDGM to a Voronoi finite-volume method (VFVM), both with upwind fluxes, demonstrating that CUDGM achieves second-order accuracy on challenging front dynamics while preserving nonnegativity, even on coarse meshes. A cut-cell formulation avoids the small-cell problem and enables robust front-tracking, with regularization ensuring stability near dry states. Numerical tests on Barenblatt, obstacle, and dambreak scenarios show that CUDGM can match or exceed VFVM accuracy at lower computational cost on coarse meshes and remains effective for realistic bathymetric data. The approach provides a practical, high-accuracy tool for flood modelling with complex terrain and wet/dry interfaces.

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.

Paper Structure

This paper contains 21 sections, 2 theorems, 51 equations, 20 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Numerical schemes
  4. Voronoi finite volume method
  5. Semi-discrete weak formulation
  6. Numerical fluxes
  7. Fully-discrete formulation
  8. Discontinuous Galerkin schemes
  9. Semi-discrete weak formulation
  10. Numerical flux and stabilization
  11. Cut-cell formulation
  12. Numerical results
  13. Barenblatt example
  14. Discussion of convergence behavior
  15. Discussion of computational cost
  16. ...and 6 more sections

Key Result

Theorem 3.1

Consider the fully discrete scheme fully_discrete_fv with homogeneous Neumann boundary conditions $j(x,t)=0$, no source or sink term $f(x,t) = 0$, and nonnegative initial condition $u^{0}_h(x_C) \geq b(x_C)$$\forall C\in\mathcal{C}_h$. Then the fully discrete solution, if it exists, satisfies

Figures (20)

  • Figure 1: The Voronoi finite volume method uses cells dual to a Delauney triangulation shown in subfigure a). The discontinuous Galerkin method uses triangular cells defined by a conforming triangular mesh shown in subfigure b).
  • Figure 2: Notation used in the definition of the flux $Q_F^{\uparrow,full}$ in Equation \ref{['eq:Q_up_full']}.
  • Figure 3: Motivation for cut-cell DG.
  • Figure 4: Illustration of cut cells and cut edges.
  • Figure 5: The analytical solution \ref{['analytical solution']} for $\vec{v} = 0$ (first row, corresponding to $b=0$) and $\vec{v} = (1/2,1/2)^T$ (second row, corresponds to $b = \vec{v}\cdot x$).
  • ...and 15 more figures

Theorems & Definitions (4)

  • Theorem 3.1: Nonnegativity of water height in the VFVM
  • proof
  • Lemma 4.1
  • proof