Moment method as a numerical solver: Challenge from shock structure problems

TL;DR

This paper reviews moment hierarchies for shock structure problems, identifies convergence issues at high Mach numbers, and introduces a new hierarchy that improves accuracy and computational efficiency for high-speed, non-equilibrium flows.

Contribution

The paper proposes a novel moment hierarchy that bridges hydrodynamic and kinetic models, enhancing numerical simulation of high Mach number shock structures.

Findings

The new method accurately predicts high Mach number shock structures.

Results converge to Boltzmann equation solutions as moments increase.

The method is extendable to three-dimensional velocity cases.

Abstract

We survey a number of moment hierarchies and test their performances in computing one-dimensional shock structures. It is found that for high Mach numbers, the moment hierarchies are either computationally expensive or hard to converge, making these methods questionable for the simulation of highly non-equilibrium flows. By examining the convergence issue of Grad's moment methods, we propose a new moment hierarchy to bridge the hydrodynamic models and the kinetic equation, allowing nonlinear moment methods to be used as a numerical tool to discretize the velocity space for high-speed flows. For the case of one-dimensional velocity, the method is formulated for odd number of moments, and it can be extended seamlessly to the three-dimensional case. Numerical tests show that the method is capable of predicting shock structures with high Mach numbers accurately, and the results converge to the solution of the Boltzmann equation as the number of moments increases. Some applications beyond the shock structure problem are also considered, indicating that the proposed method is suitable for the computation of transitional flows.