Parallel techniques for a boundary value problem with non-classic boundary conditions

Abstract

New parallel techniques are developed for obtaining approximate solutions to an initial-boundary value problem for the three-dimensional parabolic partial differential equation (PDE) with non-classic boundary conditions. While sharing some common features with the one-dimensional models, the solution of three-dimensional equations are substantially more difficult, thus some considerations are taken to be able to extend some ideas of one-dimensional case. Using a suitable transformation the solution of this problem is equivalent to the solution of two other problems. The former which is a one-dimensional non-classic boundary value problem giving the value of μ through using an unconditionally stable explicit scheme. Using this result the second problem will be changed to a classical three-dimensional diffusion equation with Neumann’s boundary conditions which will be solved numerically by using the method of lines semi-discretization approach. The space derivatives in the PDE are approximated by finite difference replacements. The solution of the resulting system of first-order linear ordinary differential equations satisfies a recurrence relation which involves a matrix exponential function. The accuracy in time is controlled by choosing several subdiagonal Pade approximants to replace this matrix exponential term. Numerical techniques are developed to compute the required solution using a splitting method, leading to algorithms for sequential and parallel implementation. The algorithms are tested on a model problem and the resulting numerical experiments are also presented. The central processor unit times needed are also reported and are compared.