Stabilizing the Maximal Entropy Moment Method for Rarefied Gas Dynamics at Single-Precision

TL;DR

This paper introduces stabilization techniques for the maximal entropy moment method, enabling accurate and efficient single-precision GPU simulations of high-speed shock waves up to Mach 10, surpassing previous double-precision capabilities.

Contribution

It proposes gauge transformations and a canonical form approach to improve the condition number and stability of MEM, allowing robust single-precision GPU simulations of rarefied gas flows.

Findings

Achieved single-precision GPU simulation of Mach 10 shock waves with 35 moments.

Improved stability of MEM through gauge transformations and canonical distribution form.

Discovered that over-refinement of spatial mesh can degrade MEM stability.

Abstract

The maximal entropy moment method (MEM) is systematic solution of the challenging problem: generating extended hydrodynamic equations valid for both dense and rarefied gases. However, simulating MEM suffers from a computational expensive and ill-conditioned maximal entropy problem. It causes numerical overflow and breakdown when the numerical precision is insufficient, especially for flows like high-speed shock waves. It also prevents modern GPUs from accelerating MEM with their enormous single floating-point precision computation power. This paper aims to stabilize MEM, making it possible to simulating very strong normal shock waves on modern GPUs at single precision. We improve the condition number of the maximal entropy problem by proposing gauge transformations, which moves not only flow fields but also hydrodynamic equations into a more optimal coordinate system. We addressed numerical overflow and breakdown in the maximal entropy problem by employing the canonical form of distribution and a modified Newton optimization method. Moreover, we discovered a counter-intuitive phenomenon that over-refined spatial mesh beyond mean free path degrades the stability of MEM. With these techniques, we accomplished single-precision GPU simulations of high speed shock wave up to Mach 10 utilizing 35 moments MEM, while previous methods only achieved Mach 4 on double-precision.