Parallel Algorithms with Local Fourier Basis

Abstract

We present a nonoverlapping domain decomposition method with local Fourier basis applied to a model problem in liquid flames. The introduction of domain decomposition techniques in this paper is for numerical and parallel efficiency purposes when one requires a large number of grid points to catch complex structures. We obtain then a high-order accurate domain decomposition method that allows us to generalize our previous work on the use of local Fourier basis to solve combustion problems with nonperiodic boundary conditions (M. Garbey and D. Tromeur-Dervout, J. Comput. Phys.145, 316 (1998)). Local Fourier basis methodology fully uses the superposition principle to split the searched solution in a numerically computed part and an analytically computed part. Our present methodology generalizes the Israeli et al. (1993, J. Sci. Comput. 8, 135) method, which applies domain decomposition with local Fourier basis to the Helmholtz's problem. In the present work, several new difficulties occur. First, the problem is unsteady and nonlinear, which makes the periodic extension delicate to construct in terms of stability and accuracy. Second, we use a streamfunction biharmonic formulation of the incompressible Navier–Stokes equation in two space dimensions: The application of domain decomposition with local Fourier basis to a fourth-order operator is more difficult to achieve than for a second-order operator. A systematic investigation of the influence of the method's parameters on the accuracy is done. A detail parallel MIMD implementation is given. We give an a priori estimate that allows the relaxation of the communication between processors for the interface problem treatment. Results on nonquasi-planar complex frontal polymerization illustrate the capability of the method.