The Boltzmann equation, a six-dimensional integro-differential equation, governs the fluid flow behavior at molecular level for a wide range of physical phenomena, including shocks, turbulence, diffusion, and non-equilibrium chemistry which are beyond the reach of continuum fluid flow modelling based on the Navier-Stokes equations. Despite Boltzmann equation’s wide applicability, its deterministic solution presents a huge computational challenge, and has been so far tractable only in simplified forms. We implement the Discontinuous Galerkin Fast Spectral (DGFS) method (Jaiswal, Alexeenko, and Hu 2019 [1]) for solving the full Boltzmann equation on streaming multi-processors. The proposed method is flexible and robust allowing: a) arbitrary un-structured geometries, b) control of spatial accuracy using high-order polynomial approximation without compromising simulation stability, c) exponential error convergence (spectral accuracy) in velocity space, and d) compact nature of DG as well as collision operator thus minimizing communication and maximizing parallel efficiency. The DG operators (for instance gradient, curl, etc), as well as the collision operator is applied in an element-local way, with flux-based element-to-element coupling. It is this locality that equips DGFS with strong parallel performance on streaming multi-processors. In the present work, we describe, devise and implement DGFS for General-Purpose Graphics Processing Units (GP-GPU). We consider the simulations of 0D spatially homogeneous, and rarefied 1D Fourier heat transfer. A speedup of approximately 10–100x, and parallel efficiency of 0.95 is demonstrated on multi-CPU/multi-GPU architectures. It is this speedup that now allows researchers to solve problems within a day that would otherwise take months on traditional CPUs. The key optimizations and techniques used to achieve these GPU performance results have been highlighted.

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