Multigrid Methods for Computational Acoustics on Vector and Parallel Computers

We consider the parabolic approximation to the three{dimensional Helmholtz equation for the acoustic pressure. The parabolic equation is semi{discretized in the range variable using an implicit scheme (e.g., Crank{Nicolson). This leads to a complex elliptic partial di erential equation that must be solved at each range step. We use a multigrid method to solve this partial di erential equation, which gives rise to matrices that are complex, symmetric (but non-Hermitean). In this sense, multigrid is an alternative to other approaches such as Krylov subspace methods and ADI. We present results for the Alliant FX2800 (a shared memorymachine based on i860 processors), and the Cray Y-MP. Our results demonstrate that multigrid maps e ectively to these supercomputer architectures, and that high performance can be achieved through the parallelizing compilers with relatively little user intervention.