A generalized fundamental solution technique for the regularized 13-moment system in rarefied gas flows

TL;DR

This work develops a generic mesh-free framework to solve linear moment systems, notably the regularized 13-moment (R13) equations for rarefied gas flows, by deriving fundamental solutions without predefined Dirac sources. It first validates the approach on Stokes equations and then constructs the R13 fundamental solution via Fourier-based symbol analysis, yielding a 2D solution composed of polyharmonic and Helmholtz components and a boundary-collocation mechanism to recover the full solution. The method is validated against an analytic R13 solution for coaxial cylinders and applied to a thermally driven flow between noncoaxial cylinders, where it is shown to outperform the FEM in convergence speed and accuracy for comparable computational effort. The study further analyzes singularity placement, boundary discretization, and conditioning, and discusses extensions to 3D problems, iterative nonlinear treatments, and broader class of moment models, highlighting the MFS as a promising alternative for complex, evolving geometries in rarefied gas dynamics.

Abstract

In this work, we explore the method of fundamental solutions (MFS) for solving the regularized 13-moment (R13) equations for rarefied monatomic gases. While previous applications of the MFS in rarefied gas flows relied on problem-specific fundamental solutions, we propose a generic approach that systematically computes the fundamental solutions for any linear moment system without predefined source terms. The generalized framework is first introduced using a simple example involving the Stokes equations, and is then extended to the R13 equations. The results obtained from the generic MFS are validated against an analytical solution for the R13 equations. Following validation, the framework is applied to the case of thermally-induced flow between two non-coaxial cylinders. Since no analytical solution exists for this case, we compare the results obtained from the MFS with those obtained from the finite element method (FEM). To further assess computational efficiency, we analyze the runtimes of the FEM and MFS. The results indicate that the MFS converges faster than the FEM and serves as a promising alternative to conventional meshing-based techniques.

Figures (10)

• Figure 1: Schematic representation for discretization of boundary points (blue disks) on the domain boundary and singularity points (red disks) outside the problem domain.
• Figure 2: Stokes' flow between two cylinders (left) and the placement of boundary nodes and singularities in the MFS (right).
• Figure 3: Schematic of the cross-section of rarefied gas confined between two coaxial cylinders where the inner cylinder is rotating anticlockwise.
• Figure 4: Variation of the speed (left panel) and temperature (right panel) in the gap between the two cylinders. The solid blue, red and black lines denote the analytic results of the R13 model along $\vartheta=0,\pi/4$ and $\pi/2$, respectively. The corresponding blue, red and black (triangle) symbols denote the results obtained from the MFS for $\mathrm{Kn}=0.5$.
• Figure 5: Variation in $L^2$ error in velocity $\epsilon_{L^2}$ and effective condition number $\kappa_{\mathrm{eff}}$ with respect to the dilation parameter $\alpha$ for different values of grid spacing $d$ and $\bm{M}=\bm B(\bm{x})^\mathsf{T}$