A collision-based hybrid method for the BGK equation

• Published in: Journal of computational physics 2024-03, Vol.501 (C), p.112784, Article 112784
• Main Authors: Shin, Minwoo, Hauck, Cory D., McClarren, Ryan G.
• Format: Article
• Language: English
• Subjects: Asymptotic-preserving; BGK equation; Discontinuous Galerkin; Discrete velocity; Gas-injection; Hybrid decomposition
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2024.112784

We apply the collision-based hybrid method introduced by Hauck and McClarren [1] to the Boltzmann equation with the BGK operator and a hyperbolic scaling. An implicit treatment of the source term is used to handle stiffness associated with the BGK operator. Although it helps the numerical scheme become stable with a large time step size, it is still not obvious to achieve the desired order of accuracy due to the relationship between the size of the spatial cell and the mean free path. Without asymptotic preserving property, a very restricted grid size is required to resolve the mean free path, which is not practical. Our approaches are based on the noncollision-collision decomposition of the BGK equation. We introduce the arbitrary order of nodal discontinuous Galerkin (DG) discretization in space with a semi-implicit time-stepping method; we employ the backward Euler time integration for the uncollided equation and the 2nd order predictor-corrector scheme for the collided equation, i.e., both source terms in uncollided and collided equations are treated implicitly and only streaming term in the collided equation is solved explicitly. This improves the computational efficiency without the complexity of the numerical implementation. Numerical results are presented for various Knudsen numbers to present the effectiveness and accuracy of our hybrid method. Also, we compare the solutions of the hybrid and non-hybrid schemes. •Enhanced numerical efficiency via hybrid (uncollided/collided equations) decomposition.•Asymptotic-preserving property.•Improved accuracy through a correction step.•Novel gas-injection problem to emphasize the efficacy of our scheme.