On high-order/low-order and micro-macro methods for implicit time-stepping of the BGK model

TL;DR

This paper introduces a combined high-order/low-order and micro-macro method for implicit time-stepping of the BGK model, improving efficiency and robustness in gas dynamics simulations, especially near equilibrium.

Contribution

It develops a novel HOLO-MM approach that accelerates implicit BGK solvers by leveraging micro-macro decomposition and distribution compression.

Findings

Both methods improve convergence speed over standard iteration.

The micro-macro approach reduces storage costs in near-equilibrium regimes.

Numerical tests demonstrate robustness and efficiency of the proposed methods.

Abstract

In this paper, a high-order/low-order (HOLO) method is combined with a micro-macro (MM) decomposition to accelerate iterative solvers in fully implicit time-stepping of the BGK equation for gas dynamics. The MM formulation represents a kinetic distribution as the sum of a local Maxwellian and a perturbation. In highly collisional regimes, the perturbation away from initial and boundary layers is small and can be compressed to reduce the overall storage cost of the distribution. The convergence behavior of the MM methods, the usual HOLO method, and the standard source iteration method is analyzed on a linear BGK model. Both the HOLO and MM methods are implemented using a discontinuous Galerkin (DG) discretization in phase space, which naturally preserves the consistency between high- and low-order models required by the HOLO approach. The accuracy and performance of these methods are compared on the Sod shock tube problem and a sudden wall heating boundary layer problem. Overall, the results demonstrate the robustness of the MM and HOLO approaches and illustrate the compression benefits enabled by the MM formulation when the kinetic distribution is near equilibrium.