Parallel iterative methods for the Helmholtz equation with exact nonreflecting boundaries

Cristian Ianculescu and Lonny L. Thompson

Department of Mechanical Engineering and Engineering Mechanics
Clemson University
Clemson, South Carolina 29634-0921

Paper IMECE2002/NCA-32744, Proc. IMECE'02, 2002 ASME International Mechanical Engineering Congress and Exposition, New Orleans, Louisiana, Nov. 17-22, 2002.

Abstract

Parallel iterative methods for fast solution of large-scale acoustic radiation and scattering problems are developed using exact Dirichlet-to-Neumann (DtN), nonreflecting boundaries. A separable elliptic nonreflecting boundary is used to efficiently model unbounded regions surrounding elongated structures. We exploit the special structure of the non-local DtN map as a low-rank update of the system matrix to efficiently compute the matrix-by-vector products found in Krylov subspace based iterative methods. For the complex non-hermitian matrices resulting from the Helmholtz equation, we use a distributed-memory parallel BICG-STAB iterative method in conjunction with a parallel Jacobi preconditioner. Domain decomposition with interface minimization was performed to ensure optimal inter-processor communication. For the architectures tested, and using the MPICH version of MPI, we show that when implemented as a low-rank update, the non-local character of the DtN map does not significantly decrease the scale up and parallel efficiency versus a purely approximate local boundary condition.