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

V. González-Tabernero, J. G. López-Salas, M. J. Castro-Díaz, J. A. García-Rodríguez

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.

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

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.
Paper Structure (30 sections, 2 theorems, 85 equations, 8 figures, 4 tables, 3 algorithms)

This paper contains 30 sections, 2 theorems, 85 equations, 8 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. IMEX LDG methods
  3. LDG space semidiscretization
  4. IMEX time discretization
  5. Boundary treatment, general technique
  6. First IMEX stage
  7. Second IMEX stage
  8. Third IMEX stage
  9. Numerical algorithms for the boundary treatment
  10. Numerical experiments
  11. One-dimensional convective heat equation, third-order test
  12. One-dimensional diffusive Burgers equation, third-order test
  13. Two-dimensional convective heat equation, third-order test \newlabelsec:exp2d0
  14. One-dimensional convective heat equation, fourth-order test \newlabelsec:expO40
  15. Boundary treatment efficiency
  16. ...and 15 more sections

Key Result

Proposition 3.1

\newlabelapproximation_tnl0 Let $H: \Omega\times [0,T] \to \Omega$ be a $C^2$ function, then for $z = u, u_x$.

Figures (8)

  • Figure 1: Examples of characteristic errors in space caused by the imposition of conventional boundary conditions on the internal stages of IMEX methods. Left: errors in the one-dimensional problem of Section \ref{['subsec:1dheat']}. Right: Contour plot of the errors for the two-dimensional problem of Section \ref{['sec:exp2d']}.
  • Figure 1: Definition of alternating numerical flux at the boundaries of interior volumes.
  • Figure 1: $|\hat{u}-u|$ for $N=160$ with the same data of Table \ref{['tb-1d-linear-source']}.
  • Figure 1: Contour plots for the spatial errors considering $N,M=80$ with the same data of Table \ref{['tb-linear2d-sourceterms']}, $t=T$.
  • Figure 2: Definition of alternating numerical flux at the boundaries of the $\textcolor{gray}{\squareneswfill}_{N-1,M-1}$ volume.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Proposition 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Theorem 3.5
  • Proof 1
  • Remark 3.6