Exponential Time Differencing Runge-Kutta Discontinuous Galerkin (ETD-RKDG) Methods for Nonlinear Degenerate Parabolic Equations

TL;DR

This paper develops high-order exponential time differencing Runge-Kutta discontinuous Galerkin methods to efficiently solve nonlinear degenerate parabolic equations with sharp fronts, improving stability and allowing larger time steps.

Contribution

It introduces a novel combination of DG spatial discretization with ETD-RK time integration for complex degenerate parabolic equations, including stability analysis and implementation on unstructured meshes.

Findings

Enhanced stability and larger time steps in simulations.

Effective capture of sharp wave fronts.

Successful application to 3D complex meshes.

Abstract

In this paper, we study high-order exponential time differencing Runge-Kutta (ETD-RK) discontinuous Galerkin (DG) methods for nonlinear degenerate parabolic equations. This class of equations exhibits hyperbolic behavior in degenerate regions and parabolic behavior in non-degenerate regions, resulting in sharp wave fronts in the solution profiles and a parabolic-type time-step restriction, $\tau \sim O(h^2)$, for explicit time integration. To address these challenges and solve such equations in complex domains, we employ DG methods with appropriate stabilizing limiters on unstructured meshes to capture the wave fronts and use ETD-RK methods for time integration to resolve the stiffness of parabolic terms. We extract the system's stiffness using the Jacobian matrix of the DG discretization for diffusion terms and adopt a nodal formulation to facilitate its computation. The algorithm is described in detail for two-dimensional triangular meshes. We also conduct a linear stability analysis in one spatial dimension and present computational results on three-dimensional simplex meshes, demonstrating significant improvements in stability and large time-step sizes.