A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem

  • Authors:

  • Frédéric Gibou;Ronald Fedkiw

  • Affiliations:

  • Mathematics Department, Stanford University, Stanford, CA 94305, USA and Computer Science Department, Stanford University, Stanford, CA 94305, USA;Computer Science Department, Stanford University, Stanford, CA 94305, USA

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2005

Quantified Score

Hi-index 31.62

Visualization

Abstract

In this paper, we first describe a fourth order accurate finite difference discretization for both the Laplace equation and the heat equation with Dirichlet boundary conditions on irregular domains. In the case of the heat equation we use an implicit discretization in time to avoid the stringent time step restrictions associated with requirements for explicit schemes. We then turn our focus to the Stefan problem and construct a third order accurate method that also includes an implicit time discretization. Multidimensional computational results are presented to demonstrate the order accuracy of these numerical methods.