Differentiable monotonicity-preserving schemes for discontinuous Galerkin methods on arbitrary meshes

TL;DR

This paper introduces a novel differentiable monotonicity-preserving interior penalty discontinuous Galerkin scheme for steady and transient transport problems on arbitrary meshes, ensuring maximum principle preservation and improved nonlinear solver convergence.

Contribution

It develops a new shock detector-based artificial diffusion scheme with a graph-Laplacian operator, enabling monotonicity and differentiability for implicit time stepping on complex meshes.

Findings

The scheme preserves maximum principles at the discrete level.

The smoothed nonlinear stabilization improves convergence of Newton solvers.

Numerical results confirm the theoretical properties and effectiveness of the method.

Abstract

This work is devoted to the design of interior penalty discontinuous Galerkin (dG) schemes that preserve maximum principles at the discrete level for the steady transport and convection-diffusion problems and the respective transient problems with implicit time integration. Monotonic schemes that combine explicit time stepping with dG space discretization are very common, but the design of such schemes for implicit time stepping is rare, and it had only been attained so far for 1D problems. The proposed scheme is based on an artificial diffusion that linearly depends on a shock detector that identifies the troublesome areas. In order to define the new shock detector, we have introduced the concept of discrete local extrema. The diffusion operator is a graph-Laplacian, instead of the more common finite element discretization of the Laplacian operator, which is essential to keep monotonicity on general meshes and in multi-dimension. The resulting nonlinear stabilization is non-smooth and nonlinear solvers can fail to converge. As a result, we propose a smoothed (twice differentiable) version of the nonlinear stabilization, which allows us to use Newton with line search nonlinear solvers and dramatically improve nonlinear convergence. A theoretical numerical analysis of the proposed schemes show that they satisfy the desired monotonicity properties. Further, the resulting operator is Lipschitz continuous and there exists at least one solution of the discrete problem, even in the non-smooth version. We provide a set of numerical results to support our findings.