Download Direct Methods for Solving the Boltzmann Equation and Study by V.V. Aristov PDF

By V.V. Aristov

This booklet is anxious with the equipment of fixing the nonlinear Boltz­ mann equation and of investigating its probabilities for describing a few aerodynamic and actual difficulties. This monograph is a sequel to the publication 'Numerical direct ideas of the kinetic Boltzmann equation' (in Russian) which used to be written with F. G. Tcheremissine and released via the Computing heart of the Russian Academy of Sciences a few years in the past. the most reasons of those books are nearly related, particularly, the examine of nonequilibrium gasoline flows at the foundation of direct integration of the kinetic equations. however, there are a few new elements within the method this subject is taken care of within the current monograph. particularly, awareness is paid to some great benefits of the Boltzmann equation as a device for contemplating nonequi­ librium, nonlinear procedures. New fields of program of the Boltzmann equation also are defined. strategies of a few difficulties are acquired with larger accuracy. Numerical strategies, akin to parallel computing, are in­ vestigated for the 1st time. The constitution and the contents of the current e-book have a few com­ mon positive factors with the monograph pointed out above, even if there are new concerns about the mathematical gear built in order that the Boltzmann equation will be utilized for brand new actual difficulties. due to this a few chapters were rewritten and checked back and a few new chapters were added.

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The collision integrals are evaluated using the Monte Carlo or the regular qudrature formulae. The interpolation in velocity space (needed to evaluate the inverse collision integrals) is performed in the same velocity lattice of nodes. We emphasize the physical properties which were taken into account when constructing the direct integration algorithms. In our schemes we tried to obtain (if possible) a clear physical analogy for any given element ofthe numerical algorithm. In fact, the discrete computational model of the Boltzmann equation was considered.

After the work by Bobylev some authors have also use the expansion in trigonometric functions. In [16] such an approach has been applied to computations on the problem of uniform relaxation in a gas of Maxwell molecules. It is not a simple problem to use such an expansion for the spatial flows owing, in particular, to an appearance of the cross derivatives in the convective term of the kinetic equation. The other possible expansion of the distribution function consists in the use of the mentioned series of Maxwellians with different parameters.

This problem of memorizing the arrays of coordinates that change at any time is analogous to memorizing the values of the distribution function for the fixed regular nodes (it is not necessary to store the coordinates of this nodes network). Of course, there is a question of successive approximation of the solution of the Boltzmann equation by the numerical solution for the scheme under consideration. The important problem consists in obtaining the system of nodes in phase space to approximate the kinetic equation in a simple and effective way.

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