Adaptive and recursive time relaxed monte carlo methods for rarefied gas dynamics

Recently a new class of Monte Carlo methods, called time relaxed Monte Carlo (TRMC), designed for the simulation of the Boltzmann equat ion close to fluid regimes has been introduced [L. Pareschi and G. Russo, SIAM J. Sci. Comput., 23 (2001), pp. 1253-1273]. A generalized Wild sum expansion of the solution is the basis of the si mulation 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 expans ion 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 us 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 [L. Pareschi and B. Wennberg, Monte Carlo Methods Appl., 7 (2001), pp. 349-358]. Several techniques can be used to truncate the recursive trees generated by the schemes without degrading 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 (direct simulation Monte Carlo) methods.