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

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Listed:
  • V. Gonz'alez-Tabernero
  • J. G. L'opez-Salas
  • M. J. Castro-D'iaz
  • J. A. Garc'ia-Rodr'iguez

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.

Suggested Citation

  • V. Gonz'alez-Tabernero & J. G. L'opez-Salas & M. J. Castro-D'iaz & J. A. Garc'ia-Rodr'iguez, 2024. "Boundary treatment for high-order IMEX Runge-Kutta local discontinuous Galerkin schemes for multidimensional nonlinear parabolic PDEs," Papers 2410.02927, arXiv.org.
    • Handle: RePEc:arx:papers:2410.02927
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    1. Wang, Haijin & Shu, Chi-Wang & Zhang, Qiang, 2016. "Stability analysis and error estimates of local discontinuous Galerkin methods with implicit–explicit time-marching for nonlinear convection–diffusion problems," Applied Mathematics and Computation, Elsevier, vol. 272(P2), pages 237-258.
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