A New Consistent Discrete-velocity Model for the Boltzmann Equation

This paper discusses the convergence of a new discrete-velocity model to the Boltz-mann equation. First the consistency of the collision integral approximation is proved. Based on this we prove the convergence of solutions for a modiied model to DiPerna-Lions solutions of the Boltz-mann equation. As a test numerical example, the solutions to the discrete problems are compared to the exact solution of the Boltzmann equation in the space homogeneous case. Introduction. The nonlinear Boltzmann equation describes the evolution of a gas which is seen as a collection of interacting particles. The model is physically relevant when the gas is rareeed and the binary collisions of particles prevail. The equation reads as follows: @f where f = f(; x; t) is the distribution function of a gas, which depends on the velocity variable , and space and time variables x; t. Also, Q(f; f) is the quadratic integral collision operator which acts on f as a function of. We leave the details to the next section and refer to the books by Cercignani, Illner and Pulvirenti 12], and Truesdell and Muncaster 34] for the physical background, mathematical theory and applications of this equation. Due to the complex structure of the collision operator and high dimension of the problem, the only way to obtain solutions of the Boltzmann equation in physically nontrivial situations is to use numerical computations. The most eecient methods for solving nonlinear kinetic problems are currently the particle methods with the two best known examples being Bird's method (Direct Simulation Monte Carlo or DSMC) 4] and the Nanbu scheme 25], later modiied by Babovsky 2]. In these methods the random dynamics of N-particle systems is modelled (the number of particles N is in practice several orders less than the actual number of molecules in a gas) and the averages of the random process realizations are taken as an approximation to the solution. The main drawback of particle methods is that the convergence of the averages is slow and the results are subject to random uctuations. If, instead of the particle approach, traditional techniques of numerical analysis of partial diierential equations are used, such as nite diierence or nite element schemes, several diiculties arise. The major one is that the numerical complexity of calculating the collision integral grows faster with the number of approximation points than is the case for particle schemes (typically O(N 2) vs. O(N)). Thus the application …