A second-order-accurate symmetric discretization of the Poisson equation on irregular domains

  • Authors:

  • Frederic Gibou;Ronald P. Fedkiw;Li-Tien Cheng;Myungjoo Kang

  • Affiliations:

  • Mathematics Department and Computer Science Department, stanford University, Stanford, California 94305;Computer Science Department, stanford University, Stanford, California 94305;Department of Mathematics, University of California San Diego, La Jolla, California 92093;Department of Mathematics, University of California Los Angeles, Los Angeles, california 90095

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2002

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Abstract

In this paper, we consider the variable coefficient Poisson equation with Dirichlet boundary conditions on an irregular domain and show that one can obtain second-order accuracy with a rather simple discretization. Moreover, since our discretization matrix is symmetric, it can be inverted rather quickly as opposed to the more complicated nonsymmetric discretization matrices found in other second-order-accurate discretizations of this problem. Multidimensional computational results are presented to demonstrate the second-order accuracy of this numerical method. In addition, we use our approach to formulate a second-order-accurate symmetric implicit time discretization of the heat equation on irregular domains. Then we briefly consider Stefan problems.