Developing extended hydrodynamics equations valid for both dense and rarefied gases remains a great challenge. A systematical solution for this challenge is the moment method describing both dense and rarefied gas behaviors with moments of gas molecule velocity distributions. Among moment methods, the maximal entropy moment method (MEM) stands out for its well-posedness and stability, which utilizes velocity distributions with maximized entropy. However, finding such distributions requires solving an ill-conditioned and computation-demanding optimization problem. This problem 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 optimization with their enormous single floating-point precision computation power. This paper aims to stabilize MEM, making it practical for simulating very strong normal shock waves on modern GPUs at single precision. We propose the gauge transformations for MEM, making the optimization less ill-conditioned. We also tackle numerical overflow and breakdown by adopting the canonical form of distribution and Newton's modified optimization method. With these techniques, we achieved a single-precision GPU simulation of a Mach 10 shock wave with 35 moments MEM, surpassing the previous double-precision results of Mach 4. Moreover, we argued that over-refined spatial mesh degrades both the accuracy and stability of MEM. Overall, this paper makes the maximal entropy moment method practical for simulating very strong normal shock waves on modern GPUs at single-precision, with significant stability improvement compared to previous methods.

翻译：发展同时适用于稠密与稀薄气体的扩展流体动力学方程仍是一大挑战。针对该挑战的系统性解决方案是矩方法，该方法通过气体分子速度分布的矩来描述稠密与稀薄气体行为。在各类矩方法中，最大熵矩方法（MEM）因其适定性与稳定性而脱颖而出，该方法采用熵最大化的速度分布。然而，求解此类分布需要解决一个病态且计算密集的优化问题。该问题在数值精度不足时（尤其针对高速激波等流动）会导致数值溢出发散，同时阻碍现代GPU利用其强大的单精度浮点运算能力加速优化。本文致力于稳定MEM，使其能够在现代GPU上以单精度模拟极强的正激波。我们提出MEM的规范变换，有效降低优化问题的病态性；通过采用分布的正则形式与牛顿改进优化方法，进一步解决数值溢出发散问题。凭借这些技术，我们成功实现了基于35矩MEM的Mach 10激波单精度GPU模拟，超越了此前双精度下Mach 4的计算结果。此外，我们论证了过细的空间网格会同时降低MEM的精度与稳定性。综上，本文使最大熵矩方法能够以单精度在现代GPU上稳定模拟极强正激波，相比传统方法稳定性显著提升。