Giovanni Russo 教授 特別講演会

In this talk we give a review of Time Relaxed Monte Carlo methods (TRMC) for the numerical solution of the Boltzmann equation of rarefied gas dynamics, and we present some recent results on the sampling from a Mac Kean graph.

TRMC scheme, in its original form, is based on writing the solution of the space homogeneous Boltzmann equation using a Wild sum expansion, truncating the Wild series by a finite sum, and replacing all the high order terms by a Maxwellian[1].

Formally the scheme is justified by some property of the Wild sum coefficients. The replacement of particles with other sampled from Maxwellians is motivated by the objective of obtaining an efficient scheme even in situations in which standard Monte Carlo approach become particularly inefficient (for example when the Knudsen number is very small, and the particle distribution is near local thermodynamical equilibrium).

Such approach, however, has several limitations, such as finite order of accuracy, and the fact that the particles replaced by a Maxwellian may not be close to equilibrium. A first attempt to extend the method to a scheme of infinite order in time has been performed by Pareschi and Wennberg[2], who constructed a scheme which recursively samples from the infinite Wild series. The truncation of the series can be performed at an arbitrarily high order. Their approach, however, does not take into account the collision history of the particles sampled from the different Wild sum coefficients.

This effect has been considered in a recent work by Pareschi and Trazzi[3], who use a recursive scheme which stops if the length of the collision history of a sampled particles (called the MacKean graph) is too long, and replaces the particle by a Maxwellian.

Recently, a new formulation of TRMC has been proposed by A.Shevyrin and G.Russo, which is based on rewriting the Wild sum expansion in terms of MacKean graphs of given level L, each of which contributes to the formation of exactly L+1 particles which have undergone at least one collision. This new approach has the advantage that each MacKean graph is exactly conservative, and therefore one can perform a simulation by sampling a certain number of independent MacKean graphs.

A detailed test on Pareshi-Wennberg recursive algorithm shows that it produces a biased average number of particles per level in the sampling from the Wild sum, while the new algorithm produces an unbiased number of particles per level.

We computed the coefficient of the new expansion, showing that in the basic algorithm (no replacement of particles by Maxwellian) the coefficients are all positive under a suitable stability condition on the ratio between time step t and Knudsen number Kn (positivity of the coefficient is essential to maintain the probabilistic interpretation of the expansion, and therefore to use such a sum as a sampling technique).

• [1] Pareschi, L. and Russo, G., Time relaxed Monte Carlo methods for the Boltzmann equation, SIAM J. Sci. Comput. 23 (2001), no. 4, 1253-1273.
• [2] Pareschi, L. and Wennberg, B., A recursive Monte Carlo method for the Boltzmann equation in the Maxwellian case, Monte Carlo and probabilistic methods for partial differential equations, Part II (Monte Carlo, 2000). Monte Carlo Methods Appl. 7 (2001), no. 3-4, 349-357.
• [3] Pareschi, L. and Trazzi, S., Numerical solution of the Boltzmann equation by time relaxed Monte Carlo (TRMC) methods, Internat. J. Numer. Methods Fluids 48 (2005), no. 9, 947-983.