Using Distributed Computers to Deterministically Approximate Higher Dimensional Convection Diffusion Equations

Abstract

A new approach to solving D > 3 spatial dimensional convection-diffusion equation on clusters of workstations is derived by exploiting the stability and scalability of the combination of a generalized D dimensional high-order compact (HOC) implicit finite difference scheme and parallelized GMRES(m). We then consider its application to multifactor Option pricing using the Black–Scholes equation and further show that an isotropic fourth order compact difference scheme is numerically stable and determine conditions under which its coefficient matrix is positive definite. The performance of GMRES(m) on distributed computers is limited by the inter-processor communication required by the matrix-vector multiplication. It is shown that the compact scheme requires approximately half the number of communications as a non-compact difference scheme of the same order of truncation error. As the dimensionality is increased, the ratio of computation that can be overlapped with communication also increases. CPU times and parallel efficiency graphs for single time step approximation of up to a 7D HOC scheme on 16 processors confirm the numerical stability constraint and demonstrate improved parallel scalability over non-compact difference schemes.