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

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

 Numerically solving partial differential equations (PDEs) remains a compelling application of supercomputing resources. The next generation of computing resources – exhibiting increased parallelism and deep memory hierarchies– provides an opportunity to rethink how to solve PDEs, especially time-dependent PDEs. Here, we consider time as an additional dimension and simultaneously solve for the unknown in large blocks of time (i.e. in 4D space-time), instead of the standard approach of sequential time-stepping. We discretize the 4D space-time domain using a mesh-free kD-tree construction that enables good parallel performance as well as on-the-fly construction of adaptive 4D meshes. To best use the 4D space-time mesh adaptivity, we invoke concepts from PDE analysis to establish rigorous posterior error estimates for a general class of PDEs. We solve canonical linear as well as non-linear PDEs (heat diffusion, advection-diffusion, and Allen-Cahn) in space-time, and illustrate the following advantages: (a) sustained scaling behavior across a larger processor count compared to sequential time-stepping approaches, (b) the ability to capture“localized” behavior in space and time using the adaptive space-time mesh, and (c) removal of anytime-stepping constraints like the Courant-Friedrichs-Lewy (CFL) condition, as well as the ability to utilize spatially varying time-steps. We believe that the algorithmic and mathematical developments along with efficient deployment on modern architectures shown in this work constitute an important step towards improving the scalability of PDE solvers on the next generation of supercomputers.