The conservative splitting method for solving Boltzmann's equation

Study of the shock wave structure by regularized Grad’s set of equations

“…The kinetic approach based on solving the Boltzmann equation is formally applicable for all degrees of rarefaction. However, in solving real applied problems by methods based on the kinetic approach, such as the Direct Simulation Monte Carlo (DSMC) method, 1 numerical solution of the Boltzmann equation, [2][3][4] and various model equations (BGK, ESM, Shakhov etc., [5][6][7], there are significant constraints associated with computational engineering capabilities. An alternative for computations of moderately rarefied flows is the use of continuum methods, which can simulate the behavior of significantly a) Electronic mail: timokhin@physics.msu.ru.…”

On conservative and entropic discrete axisymmetric Fokker-Planck operators

“…We also refer to A. V. Bobylev [7] and to P. [15] In the latter, the problem of conservation is not considered We also refer to O Larroche [16] who implemented a mass-conservmg finite volume scheme An improvement of this method was realized by D Deck & G Samba [9] yieldmg the conservation of momentum and energy and usmg a correction method exposed in V V Aristov & F G Cheremism [2] Last, we refer to M Lemou, C Buet, S Cordier & P Degond [17], for recent simulations of the 3D Fokker Planck équation, using the method descnbed m [12] and [18] In this work, the cost induced by the 3D character of the problem is decreased usmg sub-mesh methods The paper is orgamzed in the followmg way In Section 2, we first analyze the whole continuous problem in the context of the axisymmetnc geometry we show the decrease of the kmetic entropy and we charactenze the colhsional invariants In particular, we point out the crucial rôle played by the algebraic structure of the Fokker-Planck operator, which may be easily extended to the discrete case A class of discrete Fokker-Planck operators, mvolving finite différences, and preserving this algebraic structure is discussed m Section 3 Then necessary and sufficient conditions are given on the finite différence operators in order to preserve at the discrete level the solutions of the intégration step We propose in Section 4 a discrete implemented operator that preserves all the expected quanti ties and only those ones Numencal results are finally given and compared with previous computations in Section 5…”

“…with du 1 = v l ± <ft?j dv\ In formula (12), 't" = M x R* x (0, 2 n), and v = ( V, a) = (t^ , v L , a) is a cylindncal System of coordinates (the notation v^ and v ± will be precised later on), while Div and Grad dénote the divergence and gradient operators The velocity distribution ƒ = /(V) does not depend on a, yielding an operator /*(ƒ,ƒ) wfcuch also does not depend on a (we shall show this fact in Section 2) At last, &(w) is the tensor (13) Smce /-- 2 is the projection operator onto the plane orthogonal to w, <P(w) is semi-defimte positive L l w l J and lts null set is >(w) = wU (14) These two purely algebraic properties of the tensor <Z>, coupled with the fact that (-Div ) and (Grad) are adjoint operators, are precisely what we call the algebraic structure of the Fokker-Planck operator Physically speakmg, the équation (11) under considération is a model for the évolution in time t of an a-mdependent velocity distribution ƒ( t, V) of a spatially uniforrruy distributed, fully ïomsed and hot plasma, made of one species of particles which is not subrmtted to any external force Since, by use of a splitting in time algonthm, a numencal method for solving (11) also permits to simulate the évolution of a non spatially umformly distributed plasma, the mdependence with respect to the position variable is actually not restrictive Yet, the a-mdependence is usually a conséquence of some assumptions made on the spatial distribution of the plasma One of these is when the spatial distribution is only varying in one fixed direction r Introducing then r as a coordmate in this direction, the Vlasov-Fokker-Planck équation descnbmg the évolution of the plasma wntes (3, ƒ+ »" dj) (t, r, V)=P(f(t, r,. ),ƒ(*, r,. ))…”

Direct Numerical Solution Of The Boltzmann Equation

“…A few years later another method in which the collision integrals were evaluated directly at the nodes of the grid in the velocity space by Monte Carlo technique was developed by the author of this paper [2], Very soon the main deficiency of both approaches -non-fulfillment of conservation laws in the computed integrals -became evident. In [3], a splitting finite-difference scheme for the kinetic equation has been proposed, and then a special correction was developed to satisfy the conservation laws at the relaxation stage [4]. The new method showed itself much more efficient then that of [2], but the correction introduced some additional numerical viscosity, and required artificial assumptions in the case of gas mixtures [5,6].…”