Boundary treatment for high-order IMEX Runge-Kutta local discontinuous Galerkin schemes for multidimensional nonlinear parabolic PDEs

TL;DR

This paper introduces boundary treatment algorithms for high-order IMEX Runge-Kutta LDG schemes to prevent order reduction in multidimensional nonlinear parabolic PDEs with Dirichlet boundary conditions.

Contribution

The paper develops novel boundary treatment algorithms compatible with various spatial discretizations to maintain convergence order in multidimensional nonlinear PDEs.

Findings

Algorithms successfully prevent order reduction in numerical tests.

Methods recover the designed order of convergence.

Applicable to multiple spatial discretization schemes.

Abstract

In this article, we propose novel boundary treatment algorithms to avoid order reduction when implicit-explicit Runge-Kutta time discretization is used for solving convection-diffusion-reaction problems with time-dependent Di\-richlet boundary conditions. We consider Cartesian meshes and PDEs with stiff terms coming from the diffusive parts of the PDE. The algorithms treat boundary values at the implicit-explicit internal stages in the same way as the interior points. The boundary treatment strategy is designed to work with multidimensional problems with possible nonlinear advection and source terms. The proposed methods recover the designed order of convergence by numerical verification. For the spatial discretization, in this work, we consider Local Discontinuous Galerkin methods, although the developed boundary treatment algorithms can operate with other discretization schemes in space, such as Finite Differences, Finite Elements or Finite Volumes.