Time-Parallel Algorithms for Solving PDEs

A report presents additional details about a class of massively parallel algorithms for finite-difference numerical solution of time-dependent partial differential equations (PDEs). Some aspects of these algorithms were described in two previous articles in NASA Tech Briefs; namely, “Massively Parallel Computation of Electromagnetic Fields” (NPO-19453), Vol. 26, No. 5 (May 2002), page 72 and “Time-Parallel Solutions of Linear PDEs on a Supercomputer” (NPO-19385), Vol. 23, No. 12 (December 1999), page 24. These algorithms are fully parallelized in time as well as in space: this is achieved via a set of transformations based on eigenvalue/ eigenvector decompositions of matrices obtained in discretizing the PDEs. Among other things, the report discusses efficient techniques for computing these decompositions for PDEs in which the spatial part involves Laplace’s or Poisson’s equation in two-dimensional Cartesian or polar coordinates.