A meshless MUSCL method for the BGK-Boltzmann equation

TL;DR

This paper introduces a novel meshless numerical method combining MLS and ALE techniques to efficiently simulate rarefied gases with moving boundaries, achieving high-order accuracy.

Contribution

It develops a meshless MUSCL-like MLS method integrated with ALE and MOOD techniques for the BGK equation, handling complex moving boundary problems without iterative boundary condition procedures.

Findings

Achieves 4th order accuracy in 1D and 2nd order in 2D.

Effectively simulates classical test cases like cavity flow and shock tube.

Handles time-dependent boundaries and rigid objects efficiently.

Abstract

We present a numerical method for simulating rarefied gases that interact with moving boundaries and rigid bodies. The gas is described by the BGK equation in Lagrangian form and solved using an Arbitrary Lagrangian-Eulerian method, in which grid points move with the local mean velocity of the gas. The main advantage of the moving grid is that the algorithm can deal well with cases where the domain boundaries are time-dependent and the simulation domain contains rigid objects. Due to the irregular nature of the grid, we use a novel meshless MUSCL-like Moving Least Squares Method (MLS) for spatial discretisation coupled with a higher-order Implicit-Explicit Runge-Kutta method. To avoid spurious oscillations at discontinuities, we use the so-called Multi-dimensional Optimal Order Detection (MOOD) method with an adapted criterion to relax the discrete maximum property. Finally, we employ a new implementation of the boundary conditions that requires no iterative or extrapolation procedure. The method achieves fourth-order in 1D and second-order in 2D for simulations with moving boundaries. We demonstrate the method's effectiveness on classical test cases such as the driven square cavity, shear layer, and shock tube.