Discontinuous Galerkin Fast Spectral Method for Full Boltzmann Equation with General Collision Kernels: Theory, Computation, and Applications

Shashank Jaiswal, Purdue University


The Boltzmann equation, an integro-differential equation for the molecular distribution function in the physical and velocity phase space, governs the fluid flow behavior at a wide range of physical conditions, including compressible, turbulent, as well as flows involving further physics such as non-equilibrium internal energy exchange and chemical reactions. Despite its wide applicability, deterministic solution of the Boltzmann equation presents a huge computational challenge, and often the collision operator is simplified for practical reasons. In this work, a highly accurate deterministic method for the full Boltzmann equation has been introduced which couples the Runge-Kutta discontinuous Galerkin (RKDG) discretization in time and physical space, and the recently developed fast Fourier spectral method in velocity space. The novelty of this approach encompasses three aspects: first, the fast spectral method for the collision operator applies to general collision kernels with little or no practical limitations, and in order to adapt to the spatial discretization, a singular-value-decomposition based algorithm has been proposed to further reduce the cost in evaluating the collision term; second, the DG formulation employed has arbitrary order of accuracy at element-level, and has shown to be more efficient than the finite volume method; thirdly, the element-local compact nature of DG as well as our collision algorithm is amenable to effective parallelization on massively parallel architectures. The approach is applied for solving general rarefied flows with arbitrary collision kernels such as Variable Hard Sphere, Hard Sphere, and Maxwell molecular interaction. The solver has been verified against analytical Bobylev-Krook-Wu solution. Further, the standard benchmark test case of rarefied Fourier, Couette, Oscillatory-Couette, Thermal-cavity flow for different Knudsen numbers, have been studied and their results are compared against Direct Simulation Monte Carlo (DSMC) solutions. Due to multi-dimensional nature of Boltzmann-equation, the involved computations are overwhelmingly high. For study of non-trivial flow-problems, the computer-parallelization becomes necessary and unavoidable. The proposed method is therefore implemented on both Massage-Passing-Interface (MPI) based CPU-architectures, as well as GPU-architectures. The design principles behind the MPI algorithm/code implementation, have been highlighted, which allows us to express phase-space based Partial Differential Equations (PDEs) in their natural weak form directly inside the code. The idea enables straightforward development of newer codes for entire family of Boltzmann-equations such as multi-specie Boltzmann for mixtures, and Vlasov-Fokker-Plank system for plasma-modelling. Various performance evaluations such as Weak/Strong scaling including, theoretical algorithmic analysis backed with experimental performance metrics, iso-efficiency, and parallel-scalability have been discussed. Numerous micro-benchmarks involving, arbitrary collision kernels, static adaptivity of cell-size and polynomial order approximation, different time-discretization order, different quadrature types, and multi-dimensional flow cases have been studied.




Alexeenko, Purdue University.

Subject Area

Aerospace engineering|Computational physics|Computer Engineering

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