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

Journal article
Authors Stefano Trazzi
Lorenzo Pareschi
Bernt Wennberg
Published in SIAM Journal on Scientific Computing
Volume 31
Issue 2
Pages 1379-1398
ISSN 1064-8275
Publication year 2009
Published at Department of Mathematical Sciences, Mathematics
Pages 1379-1398
Language en
Keywords Boltzmann equation, Monte Carlo methods, time relaxed schemes, fluid dynamic limit, stiff systems, recursive algorithms
Subject categories Numerical analysis


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 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 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 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.

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