2,267 research outputs found
A comparative study of discrete velocity methods for low-speed rarefied gas flows
In the study of rarefied gas dynamics, the discrete velocity method (DVM) has been widely employed to solve the gas kinetic equations. Although various versions of DVM have been developed, their performance, in terms of modeling accuracy and computational efficiency, is yet to be comprehensively studied in all the flow regimes. Here, the traditional third-order time-implicit Godunov DVM (GDVM) and the recently developed discrete unified gas-kinetic scheme (DUGKS) are analysed in finding steady-state solutions of the low-speed force-driven Poiseuille and lid-driven cavity flows. With the molecular collision and free streaming being treated simultaneously, the DUGKS preserves the second-order accuracy in the spatial and temporal discretizations in all flow regimes. Towards the hydrodynamic flow regime, not only is the DUGKS faster than the GDVM when using the same spatial mesh, but also requires less spatial resolution than that of the GDVM to achieve the same numerical accuracy. From the slip to free molecular flow regimes, however, the DUGKS is slower than the GDVM, due to the complicated flux evaluation and the restrictive time step which is smaller than the maximum effective time step of the GDVM. Therefore, the DUGKS is preferable for problems involving different flow regimes, particularly when the hydrodynamic flow regime is dominant. For highly rarefied gas flows, if the steady-state solution is mainly concerned, the implicit GDVM, which can boost the convergence significantly, is a better choice
Well-balanced and asymptotic preserving schemes for kinetic models
In this paper, we propose a general framework for designing numerical schemes
that have both well-balanced (WB) and asymptotic preserving (AP) properties,
for various kinds of kinetic models. We are interested in two different
parameter regimes, 1) When the ratio between the mean free path and the
characteristic macroscopic length tends to zero, the density can be
described by (advection) diffusion type (linear or nonlinear) macroscopic
models; 2) When = O(1), the models behave like hyperbolic equations
with source terms and we are interested in their steady states. We apply the
framework to three different kinetic models: neutron transport equation and its
diffusion limit, the transport equation for chemotaxis and its Keller-Segel
limit, and grey radiative transfer equation and its nonlinear diffusion limit.
Numerical examples are given to demonstrate the properties of the schemes
General synthetic iterative scheme for nonlinear gas kinetic simulation of multi-scale rarefied gas flows
The general synthetic iteration scheme (GSIS) is extended to find the
steady-state solution of nonlinear gas kinetic equation, removing the
long-standing problems of slow convergence and requirement of ultra-fine grids
in near-continuum flows. The key ingredients of GSIS are that the gas kinetic
equation and macroscopic synthetic equations are tightly coupled, and the
constitutive relations in macroscopic synthetic equations explicitly contain
Newton's law of shear stress and Fourier's law of heat conduction. The
higher-order constitutive relations describing rarefaction effects are
calculated from the velocity distribution function, however, their
constructions are simpler than our previous work (Su et al. Journal of
Computational Physics 407 (2020) 109245) for linearized gas kinetic equations.
On the other hand, solutions of macroscopic synthetic equations are used to
inform the evolution of gas kinetic equation at the next iteration step. A
rigorous linear Fourier stability analysis in periodic system shows that the
error decay rate of GSIS can be smaller than 0.5, which means that the
deviation to steady-state solution can be reduced by 3 orders of magnitude in
10 iterations. Other important advantages of the GSIS are (i) it does not rely
on the specific form of Boltzmann collision operator and (ii) it can be solved
by sophisticated techniques in computational fluid dynamics, making it amenable
to large scale engineering applications. In this paper, the efficiency and
accuracy of GSIS is demonstrated by a number of canonical test cases in
rarefied gas dynamics.Comment: 25 pages, 17 figures; Version 3, major revision of text and
reformed/re-organized equations, added numerical analysis but numerical
results are not change
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