Reduced-Order Solution for Rarefied Gas Flow by Proper Generalised Decomposition

TL;DR

The paper tackles the computational burden of solving the high-dimensional Boltzmann/Shakhov kinetic equation for rarefied gas flows across a range of Knudsen numbers. It introduces a priori reduced-order modelling via Proper Generalised Decomposition (PGD), producing a separated, low-rank representation that yields a generalised solution (computational vademecum) for all coordinates, including the rarefaction parameter as an extra dimension. The method converts the original high-dimensional problem into a sequence of low-dimensional subproblems, significantly reducing both CPU time and memory while maintaining accuracy (e.g., flow rates within a few percent of full-rank results). Numerical results for Poiseuille and thermal-creep flows demonstrate fast construction of the general solution (around 0.26–0.41 h for the parametrised cases) and real-time access to specific solutions, with potential applicability to other particle-transport phenomena beyond gas molecules.

Abstract

Modelling rarefied gas flow via the Boltzmann equation plays a vital role in many areas. Due to the high dimensionality of this kinetic equation and the coexistence of multiple characteristic scales in the transport processes, conventional solution strategies incur prohibitively high computational costs and are inadequate for rapid response for parametric analysis and optimisation loops in engineering design simulations. This paper proposes an \textit{a priori} reduced-order method based on the proper generalised decomposition to solve the high-dimensional, parametrised Shakhov kinetic model equation. This method reduces the original problem into a few low-dimensional problem by formulating separated representations for the low-rank solution, as well as data and operators in the equation, thereby overcoming the curse of dimensionality. Furthermore, a general solution can be calculated once and for all in the whole range of the rarefaction parameter, enabling fast and multiple queries to a specific solution at any point in the parameter space. Numerical examples are presented to demonstrate the capability of the method to simulate rarefied gas flow with high accuracy and significant reduction in CPU time and memory requirements.

Figures (5)

• Figure 1: Geometry and triangular mesh for two-dimensional spatial domain. (a) Squara domain partitioning by 128 triangular elements; (b) Isosceles trapezoidal domain with a ratio of bases of 0.5 and an acute angle of 54.74°, partitioning by 128 elements; (c) Circular domain partitioning by 780 elements.
• Figure 2: Flow velocity of Poiseuille flow through a long channel with square cross section. First row (a)-(d): PGD solutions with 15 modes; second row (e)-(h): full-rank solutions; third row (i)-(l): relative errors between the low- and full-rank solutions. First column (a), (e) and (i): $\delta=100$; second column (b), (f) and (j): $\delta=10$; third column (c), (g) and (k): $\delta=1$; fourth column (d), (h) and (l): $\delta=0.1$.
• Figure 3: First four normalised PGD modes of velocity field $Y_i\times U_i/\max\{|Y_1\times U_1|\}$ of Poiseuille flow through a long channel with square cross section. First row (a)-(d): $i=1$; second row (e)-(h): $i=2$; third row (i)-(l): $i=3$; fourth row (m)-(p): $i=4$. First column (a), (e), (i) and (m): $\delta=100$; second column (b), (f), (j) and (n): $\delta=10$; third column (c), (g), (k) and (o): $\delta=1$; fourth column (d), (h), (l) and (p): $\delta=0.1$.
• Figure 4: Relative amplitudes of the modes of velocity and heat flux field: $\|\widehat{Y_iU_i}\|=\|Y_i\|\cdot\|U_i\|/\|Y_1\|\cdot\|U_1\|$ and $\|\widehat{Y_iQ_i}\|=\|Y_i\|\cdot\|Q_i\|/\|Y_1\|\cdot\|Q_1\|$. Red lines with circular markers show PGD modes. Black lines with triangular marks are the modes obtained by SVD for the full-rank solutions. First column (a) and (e): $\delta=100$; second column (b) and (f): $\delta=10$; third column (c) and (g): $\delta=1$; fourth column (d) and (h): $\delta=0.1$.
• Figure 5: (a) Dimensionless flow rate of Poiseuille flow and (b) dimensionless flow rate of thermal creep flow through long channels with trapezoidal and circular cross sections. Lines are PGD general solutions as continuous functions of rarefaction parameter $\delta$. Markers are full-rank solutions at some values of $\delta$.