Sweep-based, implicit solutions of the multidimensional BGK equation on unstructured grids
2026-08-19 • Computational Engineering, Finance, and Science
Computational Engineering, Finance, and Science
AI summaryⓘ
The authors developed a new computational method to solve the BGK kinetic equation, which describes how gases behave, using flexible grid shapes in multiple dimensions. Their approach keeps calculations accurate over time without slowing down due to tricky boundary areas. They tested their method on known problems and showed it is stable and fast, especially on GPUs, achieving big speed improvements. They also demonstrated running huge simulations on a top supercomputer using thousands of GPUs.
BGK kinetic equationnodal discontinuous Galerkin methodimplicit solverMaxwellian sourceRunge-Kutta methodGPU accelerationKokkos librarydomain decompositionparallel computingsupercomputer simulation
Authors
Thomas M. Evans, Ryan Glasby, Cory Hauck, Stefan R. Schnake, Kyle J. Schwiebert, Lawton Shoemake, Stuart Slattery
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.