Fluid Solver Independent Hybrid Methods for Multiscale Kinetic equations

TL;DR

This paper introduces a flexible hybrid method that combines Monte Carlo and macroscopic techniques to efficiently solve multiscale kinetic equations, reducing fluctuations and improving computational speed.

Contribution

It develops a form-fitting hybrid scheme compatible with various finite volume or difference methods, enhancing multiscale kinetic equation solutions.

Findings

Faster solution of multiscale fluid phenomena compared to traditional methods.

Reduced fluctuations in numerical solutions due to hybrid scheme.

Effective application to Boltzmann-BGK equation demonstrating improved performance.

Abstract

In some recent works [G. Dimarco, L. Pareschi, Hybrid multiscale methods I. Hyperbolic Relaxation Problems, Comm. Math. Sci., 1, (2006), pp. 155-177], [G. Dimarco, L. Pareschi, Hybrid multiscale methods II. Kinetic equations, SIAM Multiscale Modeling and Simulation Vol 6., No 4,pp. 1169-1197, (2008)] 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, at variance with respect to the 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 with respect to 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 less fluctuations when compared to 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.