Kernel Based High Order “Explicit” Unconditionally Stable Scheme for Nonlinear Degenerate Advection-Diffusion Equations

Andrew Christlieb, Wei Guo, Yan Jiang, Hyoseon Yang

Research output: Contribution to journalArticle

Abstract

In this paper, we present a novel numerical scheme for solving a class of nonlinear degenerate parabolic equations with non-smooth solutions. The proposed method relies on a special kernel based formulation of the solutions found in our early work on the method of lines transpose and successive convolution. In such a framework, a high order weighted essentially non-oscillatory methodology and a nonlinear filter are further employed to avoid spurious oscillations. High order accuracy in time is realized by using the high order explicit strong-stability-preserving (SSP) Runge-Kutta method. Moreover, theoretical investigations of the kernel based formulation combined with an explicit SSP method indicate that the combined scheme is unconditionally stable and up to third order accuracy. Evaluation of the kernel based approach is done with a fast O(N) summation algorithm. The new method allows for much larger time step evolution compared with other explicit schemes with the same order accuracy, leading to remarkable computational savings.

Original languageEnglish
Article number52
JournalJournal of Scientific Computing
Volume82
Issue number3
DOIs
StatePublished - Mar 1 2020

Keywords

  • High order accuracy
  • Integral solution
  • Nonlinear degenerate advection-diffusion equation
  • Unconditionally stable
  • Weighted essentially non-oscillatory methodology

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