Crank-Nicolson BGK Integrator for Multi-Scale Particle-Based Kinetic Simulations

TL;DR

This paper introduces the CN-SPBGK method, a second-order accurate particle-based solver for multi-scale gas flows that improves efficiency and accuracy across all flow regimes without extra parameters.

Contribution

The paper presents the CN-SPBGK method, a novel second-order integrator for BGK equations that enhances multi-scale kinetic simulations with better convergence and asymptotic properties.

Findings

Demonstrates improved accuracy over existing methods

Shows consistent convergence in test cases

Preserves Navier-Stokes flux asymptotically

Abstract

Solving the Bhatnagar-Gross-Krook (BGK) equation with a stochastic particle approach enables efficient and flexible simulations of flows in the transition regime, between continuum and free molecular flow. However, the usual first-order operator splitting between particle movement and relaxation imposes restrictions on the time step, causing the computational cost to increase with the gas density. The Crank-Nicolson stochastic particle BGK (CN-SPBGK) method is introduced here as an advanced particle-based kinetic solver designed for multi-scale gas flow simulations. This method integrates the BGK equation with second-order accuracy across all Knudsen number regimes without requiring additional parameters, while asymptotically preserving the Navier-Stokes flux in the continuum regime. Comparisons with pre-existing particle BGK methods are conducted on several test cases, with CN-SPBGK demonstrating more consistent convergence and accuracy.