A finite difference : boundary element scheme for solving parabolic boundary value problems and supercomputing

Abstract

The heat equation is a basic partial differential equation modelling heat conduction. Many numerical methods, such as finite-difference, finite-element, and boundary element methods, have been developed to solve the heat equation numerically. In this dissertation, we use a special form of a combination of the finite-difference and boundary-element methods to solve the heat equation, where the space variable and the time variable are discretized, respectively, by boundary elements and finite differences. The procedures may be described as follows. We use a 1-step backward Euler finite difference scheme for the time variable. This leads to a Helmholtz type equation. We then use a volume potential and a simple-layer potential to represent the solution, obtaining a boundary integral equation for the unknown simple-layer density. This boundary integral equation is then discretized by boundary elements, yielding approximate solutions of the simple-layer density, which can be further integrated to yield approximate solutions of the heat equation. Here we have taken advantage of reduction of dimensionality of the methods of boundary integral equations and boundary elements. Error estimates of the numerical scheme are analyzed, and convergence is shown. Numerical computations are carried out on the Cray Y-MP supercomputer, utilizing vectorization and parallel algorithms on the supercomputer. Some numerical results and computer graphics are also illustrated.