Acceleration Schemes of the Discrete Velocity Method: Gaseous Flows in Rectangular Microchannels

Abstract

Abstract. The convergence rate of the discrete velocity method (DVM), which has been applied extensively in the area of rarefied gas dynamics, is studied via a Fourier stability analysis. The spectral radius of the continuum form of the iteration map is found to be equal to one, which justifies the slow convergence rate of the method. Next the efficiency of the DVM is improved by introducing various acceleration schemes. The new synthetic-type schemes speed up significantly the iterative convergence rate. The spectral radius of the acceleration schemes is also studied and the so-called H 1 acceleration method is found to be the optimum one. Finally, the two-dimensional flow problem of a gas through a rectangular microchannel is solved using the new fast iterative DVM. The number of required iterations and the overall computational time are significantly reduced, providing experimental evidence of the analytic formulation. The whole approach is demonstrated using the BGK and S kinetic models.Key words. iterative methods, acceleration schemes, rarefied gas flows AMS subject classifications. 65B99, 76P05PII. S1064827502406506 Introduction. After the early work of Broadwell [4], Huang et al. [10], and Cabannes [6], the discrete velocity method (DVM) has been developed into one of the most common techniques for solving the Boltzmann equation [7,12] and simplified model equations [14,23,15] in the area of rarefied gas dynamics. The method has also been applied to solve mixture problems [20,8]. An extensive review article on internal rarefied gas flows including DVM applications has been given lately by Sharipov [16]. Very recently, new models of discrete velocity gases [5] and mixtures [9] have been introduced indicating that the method can be extended into more general models including polyatomic gases with chemical reactions.The method is based on a discretization of the velocity and space variables by choosing a suitable set of discrete velocities and by applying a consistent finite difference scheme, respectively. Then the collision integral term is approximated by an appropriate quadrature, and the resulting discrete system of equations is solved in an iterative manner. Researchers implementing the DVM are well aware, however, of its slow convergence, particularly when domains with thick subregions are considered [17]. In these cases a large number of iterations are required, and calculations are amenable to accumulated round-off error. Special attention is needed to sustain acceptable accuracy. Even more when multidimensional physical systems are examined, computational effort and time are drastically increased. These types of calculations are now needed to solve in an efficient and accurate manner fluidics applications in micro-electrical-mechanical systems and nanotechnology problems. In these applications, due to the fact that the Navier-Stokes equations are restricted by the hydrodynamic regime, kinetic-type equations and corresponding numerical approaches must

A novel experimental setup for gas microflows

“…This iteration map is terminated when the convergence criterion is fulfilled. In several occasions, it is necessary to speed up the convergence of the iteration scheme by introducing a synthetic diffusion acceleration scheme introduced in (Valougeorgis and Naris 2003). In each iteration, the system of partial differential equations 19 is solved by implementing a second-order central finite difference scheme in the physical space.…”

DSMC and R13 modeling of the adiabatic surface

“…(11), and set q y = −V x σ xy , to account for the viscous slip heating. The temperature jump corresponding to the adiabatic fully diffusive Maxwell surface is obtained from (27) 3 as…”

Switching criteria for hybrid rarefied gas flow solvers

“…The reader is referred to Valougeorgis (1988) and Valougeorgis & Naris (2003) for a detailed description of the mathematical and numerical formulation of this method (and which also deal with the application of more sophisticated model equations than the BGK approximation used here). In both simulations, 500 grid points are used.…”