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

The paper presents the implementation of a new class of massively parallel algorithms for solving certain time-dependent partial differential equations (PDEs) on massively parallel supercomputers. Such PDEs are usually solved numerically, by discretization in time and space, and by applying a time-stepping procedure to data and algorithms potentially parallelized in the spatial domain. In a radical departure from such a strictly sequential temporal paradigm, we have developed a concept of time-parallel algorithms, which allows the marching in time to be fully parallelized. This is achieved by using a set of transformations based on eigenvalue-eigenvector decomposition of the matrices involved in the discrete formalism. Our time-parallel algorithms possess a highly decoupled structure, and can therefore be efficiently implemented on emerging, massively parallel, high-performance supercomputers, with a minimum of communication and synchronization overhead. We have successfully carried out a proof-of-concept demonstration of the basic ideas using a two-dimensional heat equation example implemented on the Intel Touchstone Delta supercomputer. Our results indicate that linear, and even superlinear, speed-up can be achieved and maintained for a very large number of processor nodes.