Time-parallel solution of linear partial differential equations on the Intel Touchstone Delta supercomputer

Evolutionary partial differential actuations are usually solved by discretization in time and space, and by applying a marching in time procedure to data and algorithms potentially --2 parallelized in the spatial domain. In a cleparture from such a strictly. sequential temporal paradigm, we have developed the concept of tinle~~rgll~l__~~orithms, which allow -_..— the marching in time to be fully parallelized. This is achieved by using a transformation based on the eigenvalue-eigenvector decomposition of the matrices resulting from the ,-....– -– -..—-. discretization process. Since these matrices are involved in the time stepping iterations, the resulting diagonalization yields ‘(time parallel” algorithms, i.e., algorithms that possess a highly decoupled temporal structure, and hence can be efficiently implemented on emerging massively parallel MIMD architectures with a minimum of communication and synchronization overhead. Specifically, our algorithms have been implemented on the Intel Touchstone Delta supercomputer. We have illustrated our approach on a two-dimensional heat equation, and have demonstrated that a linear speedup can be achieved and maintained, even for a very large number of processor nodes.