A Fourth Order Accurate Discretization for the Laplace and Heat Equations on Arbitrary Domains, with Applications to the Stefan Problem

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 time discretization to avoid the stringent time step restrictions associated with 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. ∗Research supported in part by an ONR YIP and PECASE award (N00014-01-1-0620), a Packard Foundation Fellowship, a Sloan Research Fellowship, ONR N00014-03-1-0071, ONR N00014-02-1-0720, NSF DMS-0106694 and NSF ACI-0323866. In addition, the first author was supported in part by an NSF postdoctoral fellowship (DMS-0102029). †Mathematics Department & Computer Science Department, Stanford University, Stanford, CA 94305. ‡Computer Science Department, Stanford University, Stanford, CA 94305.