Stochastic models for Boltzmann-type equation

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Stochastic models for Boltzmann-type equation

Collaborator: W. Wagner, I. Matheis

Cooperation with: H. Babovsky (Technische Universität Ilmenau), S.M. Ermakov (St. Petersburg University, Russia), I.M.Gamba (University of Texas, Austin, USA), M. Kraft (University of Cambridge, UK), C. Lécot (Université de Savoie, Chambéry, France), J. Norris (University of Cambridge, UK), S. Rjasanow (Universität des Saarlandes, Saarbrücken), K.K. Sabelfeld (WIAS: Research Group 6)

Supported by: DFG: ``Effektive Steuerung von stochastischen Partikelverfahren für Strömungen in verdünnten Gasen'' (Effective control of stochastic particle methods for rarefied gas flow); EU (INTAS): ``Random walk models for the footprint problem in the turbulent atmosphere''

Description:

Rarefied gas flows play an important role in applications like aerospace design (space shuttle reentry), vacuum engineering (material processing, pumps), or, more recently, nanotechnology. Mathematically such flows are described (in the simplest case of a monatomic gas) by the Boltzmann equation

$\displaystyle {\frac{{\partial}}{{\partial t}}}$ f (t, x, v)+(v, $\displaystyle \nabla_{x}^{}$ ) f (t, x, v)=
		$\displaystyle \int_{{{\cal R}^3}}^{}$ dw $\displaystyle \int_{{{\cal S}^2}}^{}$ de B(v, w, e) $\displaystyle \Big[$ f (t, x, v^) f (t, x, w^) - f (t, x, v) f (t, x, w) $\displaystyle \Big]$ ,	(1)

where

v^* = v + e (e, w - v) , w^* = w + e (e, v - w) .

(2)

The solution f (t, x, v) represents the relative amount of gas molecules with velocity v at position x and time t . The quadratic nonlinearity in (1) corresponds to the pairwise interaction between gas particles, which consists in the change of velocities of two particles according to (2). Here $\cal {S}$ ² denotes the unit sphere in the Euclidean space $\cal {R}$ ³ , and B is called the collision kernel, containing information about the assumed microscopic interaction potential.

A nonlinear equation of similar structure as equation (1) is Smoluchowski's coagulation equation

$\displaystyle {\frac{{\partial}}{{\partial t}}}$ c(t, x)

$\displaystyle {\frac{{1}}{{2}}}$ $\displaystyle \sum_{{y=1}}^{{x-1}}$ K(x - y, y) c(t, x - y) c(t, y) - $\displaystyle \sum_{{y=1}}^{{\infty}}$ K(x, y) c(t, x) c(t, y) ,

(3)

where t $\ge$ 0 and x = 1, 2,... . It describes the time evolution of the average concentration of particles of a given size in some spatially homogeneous physical system. The concentration of particles of size x increases as a result of coagulation of particles of sizes x - y and y . It decreases if particles of size x merge with any other particles. The intensity of the process is governed by the (non-negative and symmetric) coagulation kernel K representing properties of the physical medium. The phenomenon of the coagulation occurs in a wide range of applications, e.g., in physics (aggregation of colloidal particles, growth of gas bubbles), meteorology (merging of drops in atmospheric clouds, aerosol transport), chemistry (reacting polymers, soot formation), and astrophysics (formation of stars and planets).

The purpose of the project is to study the relationship between stochastic interacting particle systems and solutions of equations of type (1) or (3). On the one hand, results on the asymptotic behavior of the particle system (when the number of particles increases) provide insight into properties of the solution. On the other hand, appropriate stochastic particle systems are used for the numerical treatment of the macroscopic equation. Stochastic numerical methods provide results that are subject to random fluctuations. Thus, the construction of algorithms with reduced fluctuations is an important issue (variance reduction problem).

In recent years a new approach to the variance reduction problem for the Boltzmann equation (1), called SWPM (stochastic weighted particle method), has been developed. The new method, which uses a system of weighted particles, has been successfully applied to situations with strong density gradients (cf. [1]). In [2] convergence of SWPM was studied. First the method was extended by introducing new stochastic reduction procedures, in order to control the number of simulation particles. Then, under rather general conditions, convergence to the solution of the Boltzmann equation was proved. Finally, numerical experiments illustrated both convergence and considerable variance reduction for the specific problem of calculating tails of the velocity distribution. The assumptions of the convergence theorem were significantly weakened in [3], in order to cover deterministic reduction procedures. First steps towards a new field of applications were taken in [4]. A new stochastic numerical algorithm was derived for the Boltzmann equation for rarefied granular flows, where collisions between particles are inelastic.

The topic of studying stochastic models for Boltzmann-type equations has attracted much interest in recent years. An interesting direction of research is the consideration of more physical effects. In [5] the stochastic approach to nonlinear kinetic equations (without gradient terms) has been presented in a unifying general framework, which covers many interactions important in applications, like coagulation, fragmentation, inelastic collisions, as well as source and efflux terms. Conditions for the existence of corresponding stochastic particle systems in the sense of regularity (non-explosion) of a jump process with unbounded intensity are provided. Using an appropriate space of measure-valued functions, relative compactness of the sequence of processes is proved, and the weak limits are characterized in terms of solutions to the nonlinear equation. As a particular application, existence theorems for Smoluchowski's coagulation equation with fragmentation, efflux and source terms, and for the Boltzmann equation with dissipative collisions are derived. Some results concerning clusters containing several chemical species are presented in [6], [7]. For sufficiently fast increasing coagulation kernels, there exists the phenomenon of gelation. At the level of the macroscopic equation (3), the gelation effect is represented by a loss of mass of the solution. An appropriate interpretation of this phenomenon in terms of stochastic particle systems is of both theoretical and practical interest. Some conjectures based on detailed numerical observations have been stated in [8].

References:

S. RJASANOW, W. WAGNER, Simulation of rare events by the stochastic weighted particle method for the Boltzmann equation, Math. Comput. Modelling, 33 (2002), pp. 907-926.
I. MATHEIS, W. WAGNER, Convergence of the stochastic weighted particle method for the Boltzmann equation, SIAM J. Sci. Comput., 24 (2003), pp. 1589-1609.
W. WAGNER, Stochastic models and Monte Carlo algorithms for Boltzmann type equations, WIAS Preprint no. 831, 2003.
I.M. GAMBA, S. RJASANOW, W. WAGNER, Direct simulation of the uniformly heated granular Boltzmann equation, WIAS Preprint no. 834, 2003.
A. EIBECK, W. WAGNER,
Stochastic interacting particle systems and nonlinear kinetic equations, Ann. Appl. Probab., 13 (2003), pp. 845-889.
M. KRAFT, W. WAGNER, Numerical study of a stochastic particle method for homogeneous gas-phase reactions, Comput. Math. Appl., 45 (2003), pp. 329-349.
, An improved stochastic algorithm for temperature-dependent homogeneous gas phase reactions, J. Comput. Phys., 185 (2003), pp. 139-157.
W. WAGNER, Stochastic, analytic and numerical aspects of coagulation processes, Math. Comput. Simulation, 62 (2003), pp. 265-275.