Domain decomposition methods for the parallel computation of reacting flows

Domain decomposition is a natural route to parallel computing for partial differential equation solvers. In this procedure, subdomains of which the original domain of definition is comprised are assigned to independent processors at the price of periodic coordination between processors to compute global parameters and maintain the requisite degree of continuity of the solution at the subdomain interfaces. In the domain-decomposed solution of steady multidimensional systems of PDEs by finite difference methods using a pseudo-transient version of Newton iteration, the only portion of the computation which generally stands in the way of efficient parallelization is the solution of the large, sparse linear systems arising at each Newton step. For some Jacobian matrices drawn from an actual two-dimensional reacting flow problem, we make comparisons between relaxation-based linear solvers and also preconditioned iterative methods of Conjugate Gradient and Chebyshev type, focusing attention on both iteration count and global inner product count. The generalized minimum residual method with block-ILU preconditioning is judged the best serial method among those considered, and parallel numerical experiments on the Encore Multimax demostrate for it approximately 10-fold speedup on 16 processsors. The three special features of reacting flow models in relation to these linear systems are: the possibly large number of degrees of freedom per gridpoint, the dominance of dense intra-point source-term coupling over inter-point convective-diffusive coupling throughout significant portions of the flow-field and strong nonlinearities which restrict the time step to small values (independent of linear algebraic considerations) throughout significant portions of the iteration history. Though these features are exploited to advantage herein, many aspects of the paper are applicable to the modeling of general convective-diffusive systems.