Fluid Solver Independent Hybrid Methods for Multiscale Kinetic Equations

Abstract

In some recent works [G. Dimarco and L. Pareschi, Comm. Math. Sci., 1 (2006), pp. 155-177; Multiscale Model. Simul., 6 (2008), pp. 1169-1197] we developed a general framework for the construction of hybrid algorithms which are able to face efficiently the multiscale nature of some hyperbolic and kinetic problems. Here, in contrast to previous methods, we construct a method form-fitting to any type of finite volume or finite difference scheme for the reduced equilibrium system. Thanks to the coupling of Monte Carlo techniques for the solution of the kinetic equations with macroscopic methods for the limiting fluid equations, we show how it is possible to solve multiscale fluid dynamic phenomena faster than using traditional deterministic/stochastic methods for the full kinetic equations. In addition, due to the hybrid nature of the schemes, the numerical solution is affected by fewer fluctuations than are standard Monte Carlo schemes. Applications to the Boltzmann-BGK equation are presented to show the performance of the new methods in comparison with classical approaches used in the simulation of kinetic equations.