An efficient numerical method for solving the Boltzmann equation in multidimensions

TL;DR

This paper extends the Fast Kinetic Scheme to solve the multidimensional Boltzmann equation efficiently, combining innovative Lagrangian transport and spectral collision methods, enabling high-dimensional rarefied gas flow simulations.

Contribution

The paper introduces an extended FKS method for the Boltzmann equation with parallelization, achieving efficient 3D simulations and providing benchmarks for future numerical comparisons.

Findings

Successfully simulated 3D unsteady flows with the Boltzmann equation.

Achieved computational efficiency and detailed resource analysis.

Provided benchmark results for future method comparisons.

Abstract

In this paper we deal with the extension of the Fast Kinetic Scheme (FKS) [J. Comput. Phys., Vol. 255, 2013, pp 680-698] originally constructed for solving the BGK equation, to the more challenging case of the Boltzmann equation. The scheme combines a robust and fast method for treating the transport part based on an innovative Lagrangian technique supplemented with fast spectral schemes to treat the collisional operator by means of an operator splitting approach. This approach along with several implementation features related to the parallelization of the algorithm permits to construct an efficient simulation tool which is numerically tested against exact and reference solutions on classical problems arising in rarefied gas dynamic. We present results up to the $3$D$\times 3$D case for unsteady flows for the Variable Hard Sphere model which may serve as benchmark for future comparisons between different numerical methods for solving the multidimensional Boltzmann equation. For this reason, we also provide for each problem studied details on the computational cost and memory consumption as well as comparisons with the BGK model or the limit model of compressible Euler equations.