A Parallel Exponential Integrator for Large-Scale Discretizations of Advection-Diffusion Models

We propose a parallel implementation of the ReLPM (Real Leja Points Method) for the exponential integration of large sparse systems of ODEs, generated by Finite Element discretizations of 3D advectiondiffusion models. The performance of our parallel exponential integrator is compared with that of a parallelized Crank-Nicolson (CN) integrator, where the local linear solver is a parallel BiCGstab accelerated with the approximate inverse preconditioner FSAI. We developed message passing codes written in Fortran 90 and using the MPI standard. Results on SP5 and CLX machines show that the parallel efficiency raised by the two algorithms is comparable. ReLPM turns out to be from 3 to 5 times faster than CN in solving realistic advection-diffusion problems, depending on the number of processors employed. 1 Finite Element Discretization of the AdvectionDiffusion Model We consider the classical evolutionary advection-diffusion problem    ∂c ∂t = div(D∇c)− div(cv) + φ x ∈ Ω, t > 0 c(x, 0) = c0(x), x ∈ Ω; c(x, t) = gD(x, t), x ∈ ΓD; 〈D∇c(x, t), ν〉 = gN(x, t), x ∈ ΓN; t > 0 (1) with mixed Dirichlet and Neumann boundary conditions on ΓD ∪ ΓN = ∂Ω, Ω ⊂ R. Equation (1) represents, e.g., a simplified model for solute transport in groundwater flow (advection-dispersion), where c is the solute concentration, D the hydrodynamic dispersion tensor, Dij = αT|v|δij + (αL − αT)vivj/|v|, 1 ≤ i, j ≤ d, v the average linear velocity of groundwater flow and φ the source. The standard Galerkin Finite Element (FE) discretization of (1) with nodes {xi}i=1 and linear basis functions gives a large scale linear system of ODEs like { P ċ = Hc + b, t > 0