Boundary treatment of linear multistep methods for hyperbolic conservation laws

Hujian Zuo, Weifeng Zhao (Lead / Corresponding author), Ping Lin

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When using high-order schemes to solve hyperbolic conservation laws in bounded domains, it is necessary to properly treat boundary conditions so that the overall accuracy and stability are maintained. In [1, 2] a finite difference boundary treatment method is proposed for Runge-Kutta methods of hyperbolic conservation laws. The method combines an inverse Lax-Wendroff procedure and a WENO type extrapolation to achieve desired accuracy and stability. In this paper, we further develop the boundary treatment method for high-order linear multistep methods (LMMs) of hyperbolic conservation laws. We test the method through both 1D and 2D benchmark numerical examples for two third-order LMMs, one with a constant time step and the other with a variable time step. Numerical examples show expected high order accuracy and excellent stability. In addition, the approach in [3] may be adopted to deal with an exceptional case where eigenvalues of the flux Jacobian matrix change signs at the boundary. These results demonstrate that the combined boundary treatment method works very well for LMMs of hyperbolic conservation laws.

Original languageEnglish
Article number127079
Number of pages12
JournalApplied Mathematics and Computation
Early online date19 Mar 2022
Publication statusPublished - 15 Jul 2022


  • Boundary treatment
  • Hyperbolic conservation laws
  • Inverse lax-Wendroff
  • Linear multistep methods

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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