Parallel LOD methods for second order time dependent PDEs

Abstract

Time dependent problems in Partial Differential Equations (PDEs) are often solved by the Method Of Lines (MOL). In many important instances, the exact solution of the resulting system of Ordinary Differential Equations (ODEs) satisfies recurrence relations involving the matrix exponential function. This function is approximated by a new type of rational function possessing real and distinct poles which, consequently, readily admits a partial fraction expansion thereby allowing the distribution of the work in solving the corresponding linear algebraic systems in essentially Backward Euler-like solves on concurrent processors. Locally One Dimensional (LOD) splitting methods with enhanced parallelization are developed for multispace problems utilizing Strang-like splitting techniques. The use of rational approximants with distinct real poles in the temporal direction, in collusion with splitting techniques in the spatial directions, creates the potential for efficient coarse grain time-stepping parallel algorithms on MIMD machines. The resulting parallel algorithms possess appropriate stability properties, and are implemented on various parabolic and hyperbolic PDEs from the literature in higher space dimensional problems.