2021/07/23 by Walter Boscheri, Giacomo Dimarco, Boscheri, Walter +1
Engineering · Mathematics · #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Gas Dynamics and Kinetic Theory #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2107.11101
openalex publication_date 2021/07/23 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
Deterministic solutions of the Boltzmann equation represent a real challenge\ndue to the enormous computational effort which is required to produce such\nsimulations and often stochastic methods such as Direct Simulation Monte Carlo\n(DSMC) are used instead due to their lower computational cost. In this work, we\nshow that combining different technologies for the discretization of the\nvelocity space and of the physical space coupled with suitable time integration\ntechniques, it is possible to compute very precise deterministic approximate\nsolutions of the Boltzmann model in different regimes, from extremely rarefied\nto dense fluids, with CFL conditions only driven by the hyperbolic transport\nterm. To that aim, we develop modal Discontinuous Galerkin (DG)\nImplicit-Explicit Runge Kutta schemes (DG-IMEX-RK) and Implicit-Explicit Linear\nMultistep Methods based on Backward-Finite-Differences (DG-IMEX-BDF) for\nsolving the Boltzmann model on multidimensional unstructured meshes. The\nsolution of the Boltzmann collision operator is obtained through fast spectral\nmethods, while the transport term in the governing equations is discretized\nrelying on an explicit shock-capturing DG method on polygonal tessellations in\nthe physical space. A novel class of WENO-type limiters, based on a shifting of\nthe moments of inertia for each zone of the mesh, is used to control spurious\noscillations of the DG solution across discontinuities. The order of\nconvergence is numerically measured for different regimes and found to agree\nwith the theoretical findings. The new methods are validated considering\ntwo-dimensional benchmark test cases typically used in the fluid dynamics\ncommunity. A prototype engineering problem consisting of a supersonic flow\naround a NACA 0012 airfoil with space-time-dependent boundary conditions is\nalso presented for which the pressure coefficients are measured.\n