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

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.

💡 Research Summary

This paper introduces a generalized Method of Fundamental Solutions (MFS) that can be applied to any linear first‑order partial‑differential system, specifically targeting the regularized 13‑moment (R13) equations used for modeling rarefied monatomic gas flows. Traditional MFS implementations for rarefied gas dynamics have required problem‑specific Dirac‑delta source terms to be manually prescribed, followed by the derivation of corresponding fundamental solutions. Such a procedure limits the method’s extensibility to new or more complex models because the appropriate source terms are not obvious a priori.

The authors overcome this limitation by formulating the governing equations in matrix form

 Aₓ ∂ₓU + A_y ∂_yU + P U = S δ(r),

where U∈ℝᴺ is the vector of unknown fields, Aₓ, A_y and P are constant coefficient matrices, and S is a constant forcing vector. Applying a Fourier transform yields

 (i kₓAₓ + i k_yA_y + P) Û = S δ̂ .

The matrix (i kₓAₓ + i k_yA_y + P) is inverted by separating its adjugate matrix and its determinant s(k), which is identified as the symbol of the differential operator. The key observation is that

 s(k) Û = A(k) S δ̂

implies, after inverse Fourier transformation, that the scalar fundamental solution Φ satisfies

 s(∇) Φ = δ .

If s(∇) can be factorized into known operators (Laplace, Helmholtz, poly‑harmonic, etc.), Φ can be constructed analytically using partial‑fraction decomposition and known inverse transforms. The full vectorial fundamental solution for the original system is then

 U(r) = A(∇)