Adaptive and Recursive Time Relaxed Monte Carlo methods for rarefied gas dynamics

Abstract

Recently a new class of Monte Carlo methods, called Time Relaxed Monte Carlo (TRMC), designed for the simulation of the Boltzmann equation close to fluid regimes have been introduced. A generalized Wild sum expansion of the solution is at the basis of the simulation schemes. After a splitting of the equation the time discretization of the collision step is obtained from the Wild sum expansion of the solution by replacing high order terms in the expansion with the equilibrium Maxwellian distribution; in this way speed up of the methods close to fluid regimes is obtained by efficiently thermalizing particles close to the equilibrium state. In this work we present an improvement of such methods which allows to obtain an effective uniform accuracy in time without any restriction on the time step and subsequent increase of the computational cost. The main ingredient of the new algorithms is recursivity. Several techniques can be used to truncate the recursive trees generated by the schemes without deteriorating the accuracy of the numerical solution. Techniques based on adaptive strategies are presented. Numerical results emphasize the gain of efficiency of the present simulation schemes with respect to standard DSMC methods.