Solving PDEs in Space-Time: 4D Tree-Based Adaptivity, Mesh-Free and Matrix-Free Approaches

Authors: Masado Ishii (University of Utah), Milinda Fernando (University of Utah), Kumar Saurabh (Iowa State University), Biswajit Khara (Iowa State University), Baskar Ganapathysubramanian (Iowa State University), Hari Sundar (University of Utah)

Abstract: We present a kD tree-based parallel strategy for solving with a spacetime-adaptive approach a general class of partial differential equations. This approach is primarily motivated by the necessity of designing computational methodologies that can scale to leverage the availability of very large computing clusters (exascale and beyond). For evolution problems, the standard approach of decomposing the spatial domain is a powerful paradigm of parallelization. However, for a fixed spatial discretization, the efficiency of purely spatial domain decomposition degrades substantially beyond a threshold. To overcome this, we consider the time domain as an additional dimension and simultaneously solve for blocks of time (spacetime), instead of the standard approach of sequential time-stepping. Spacetime discretization includes natural incorporation a posterior error, full-time solution history, and removal of time-stepping constraints. We present scalable algorithms to perform matrix and matrix-free computations on KD trees, and show scalability across 32K cores in Titan.


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