Time-Parallel Solutions of Linear PDEs on a Supercomputer

A paper describes the mathematical basis and some applications of a class of massively parallel algorithms for finite-difference numerical solution of some time-dependent partial differential equations (PDEs) on massively parallel supercomputers. In a radical departure from the traditional spatially parallel but temporally sequential approach to solution of finite-difference equations, the algorithms described in the paper 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. The resulting time-parallel algorithms exhibit highly de-coupled structures, and can therefore be efficiently implemented on emerging, massively parallel, high-performance supercomputers.