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arXiv · 2608.19150

Sweep-based, implicit solutions of the multidimensional BGK equation on unstructured grids

Thomas M. Evans·Ryan Glasby·Cory Hauck·Stefan R. Schnake·Kyle J. Schwiebert·Lawton Shoemake·Stuart Slattery
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Abstract

We present a nodal discontinuous Galerkin method for solving the Bhatnagar-Gross-Krook (BGK) kinetic equation on multi-dimensional, unstructured grids. The method uses implicit, sweep-based solvers and a moment-preserving projection of the Maxwellian source to enable high-order accuracy in time while avoiding restrictive time steps imposed by boundary layers and other geometry-induced features. We verify that the method is correct in the continuum limit by comparing to closed-form and high-order solutions of the Sod shock problem on 2 and 3D unstructured grids. Linear L2 stability is demonstrated for a B-stable diagonally implicit Runge-Kutta method of third order. The solver uses a hybrid parallel scheme based on spatial domain decomposition with local sweeps performed on CPU and GPU hardware. Platform-portability is demonstrated through the development of new GPU-friendly, graph-based sweep algorithms that are implemented using the Kokkos performance portability library and achieve greater than 20 times speedup on NVIDIA H100 GPUs compared to 64-core AMD EPYC 9654 CPUs. Finally, we show results on the Frontier supercomputer at the Oak Ridge Leadership Computing Facility for a boundary value problem with 2.77 trillion phase space degrees of freedom that executed on 1536 nodes utilizing 6144 AMD MI250X GPUs.

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Thomas M. Evans, Ryan Glasby, Cory Hauck, Stefan R. Schnake, Kyle J. Schwiebert, Lawton Shoemake, Stuart Slattery. 2026-08-19. Sweep-based, implicit solutions of the multidimensional BGK equation on unstructured grids. https://arxiv.org/abs/2608.19150

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