Parallel Space-Time Approaches for the Numerical Solution of Partial Differential Equations

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Mathematically, many models of interest to engineers, economists and scientists are written as partial differential equations (PDEs). Except for certain idealized situations, the PDEs which result are not possible to solve analytically. Instead we rely on numerical approximations. I am particularly interested in the study of efficient implementations and analysis of adaptive algorithms for the solution of time-dependent PDEs in two or three spatial dimensions whose solutions exhibit large solution variation, singularity formation or moving fronts. We study methods which attempt to obtain a solution efficiently by concentrating computational effort (in both space and time) in regions where the solution has interesting but difficult to track behavior. The strategy works by using moving spatial meshes to track interesting features of the solution forward in time. Moreover, there is an opportunity and motivation to study algorithms designed to take advantage of the continually evolving computing hardware - readily available commodity clusters with hundreds or thousands of cores, hybrid CPU-GPU systems and even desktop machines with 4-24 cores. We propose mapping the solution of time dependent PDEs to multi-core environments by dividing the large problem into small pieces computed on individual cores and recombined to give a solution of the original problem using domain decomposition (DD) algorithms. Such a strategy will be used for both the generation of the adaptive grids, as well as the solution of the physical PDE. Small scale parallelism in time may be added by computing simultaneous predictions and corrections. Ultimately, we will provide a new, theoretically based, modular platform for the parallel adaptive solution of time dependent PDEs suitable for existing and emerging HPC hardware. This research program provides a route to impact for moving mesh methods, providing a software tool for computational scientists for the solution of complex problems. Theoretically it will enhance our knowledge of the behaviour of DD algorithms for nonlinear problems. Finally, it will provide HQP with mathematical expertise and computational competency - transferable skills highly sought by employers.