Search

Continuum mechanics and thermodynamics none, none (2022). doi:10.1007/s00161-022-01167-8, Published by Springer, Berlin ; Heidelberg

Journal of scientific computing 95(3), 78 (2023). doi:10.1007/s10915-023-02168-8, Published by Springer Science + Business Media B.V., New York, NY [u.a.]

Section of the proceedings of FEniCS 2021

A jump-diffusion process along with a particle scheme is devised as an accurate and efficient particle solution to the Boltzmann equation. The proposed process (hereafter Gamma-Boltzmann model) is devised to match the evolution of all moments up to the heat fluxes while attaining the correct Prandtl number of 2/3 for monatomic gas with Maxwellian molecular potential. This approximation model is not subject to issues associated with the previously developed Fokker-Planck (FP) based models; such as having wrong Prandtl number, limited applicability, or requiring estimation of higher-order moments. An efficient particle solution to the proposed Gamma-Boltzmann model is devised and compared computationally to the direct simulation Monte Carlo and the cubic FP model [M. H. Gorji, M. Torrilhon, and P. Jenny, J. Fluid Mech. 680 (2011): 574-601] in several test cases including Couette flow and lid-driven cavity. The simulation results indicate that the Gamma-Boltzmann model yields a good approximation of the Boltzmann equation, provides a more accurate solution compared to the cubic FP in the limit of a low number of particles, and remains computationally feasible even in dense regimes.

A solution to the linear Boltzmann equation satisfies an energy bound, which reflects a natural fact: The energy of particles in a finite volume is bounded in time by the energy of particles initially occupying the volume augmented by the energy transported into the volume by particles entering the volume over time. In this paper, we present boundary conditions (BCs) for the spherical harmonic (PN) approximation, which ensure that this fundamental energy bound is satisfied by the PN approximation. Our BCs are compatible with the characteristic waves of PN equations and determine the incoming waves uniquely. Both, energy bound and compatibility, are shown based on abstract formulations of PN equations and BCs to isolate the necessary structures and properties. The BCs are derived from a Marshak type formulation of BC and base on a non-classical even/odd-classification of spherical harmonic functions and a stabilization step, which is similar to the truncation of the series expansion in the PN method. We show that summation by parts (SBP) finite differences on staggered grids in space and the method of simultaneous approximation terms (SAT) allows to maintain the energy bound also on the semi-discrete level., Comment: 26 pages, 17 figures

In (Commun Pure Appl Math 2(4):331-407, 1949), Grad proposed a Hermite series expansion for approximating solutions to kinetic equations that have an unbounded velocity space. However, for initial boundary value problems, poorly imposed boundary conditions lead to instabilities in Grad's Hermite expansion, which could result in non-converging solutions. For linear kinetic equations, a method for posing stable boundary conditions was recently proposed for (formally) arbitrary order Hermite approximations. In the present work, we study $L^2$-convergence of these stable Hermite approximations, and prove explicit convergence rates under suitable regularity assumptions on the exact solution. We confirm the presented convergence rates through numerical experiments involving the linearised-BGK equation of rarefied gas dynamics.

In this paper we extend the recently developed third-order limiter function $H_{3\text{L}}^{(c)}$ [J. Sci. Comput., (2016), 68(2), pp.~624--652] to make it applicable for more elaborate test cases in the context of finite volume schemes. This work covers the generalization to non-uniform grids in one and two space dimensions, as well as two-dimensional Cartesian grids with adaptive mesh refinement (AMR). The extension to 2D is obtained by the common approach of dimensional splitting. In order to apply this technique without loss of third-order accuracy, the order-fix developed by Buchm\"uller and Helzel [J. Sci. Comput., (2014), 61(2), pp.~343--368] is incorporated into the scheme. Several numerical examples on different grid configurations show that the limiter function $H_{3\text{L}}^{(c)}$ maintains the optimal third-order accuracy on smooth profiles and avoids oscillations in case of discontinuous solutions.

Journal of computational physics 430, 110034 (2021). doi:10.1016/j.jcp.2020.110034, Published by Elsevier, Amsterdam