Monte Carlo Methods in Mechanics of Fluid and Gas
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NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI World ScientificO. M. BelotserkovskiiRussian Academy of Sciences, RussiaY. I. KhlopkovRussian Academy of Natural Sciences, RussiaMonte Carlo Methods in Mechanics of Fluid and Gas
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PrefaceOur dedication to the pioneers of the use of Monte Carlo methods in mechan-ics of fluid and gas in Russia Vladimir Alexandrovich Perepukhov and VitaliyEvgenjevichYanitskii.Thebeginningofthethirdmillenniumischaracterizedbytheglobaluniquenessof the human civilization. The possibilities of humanity in energetic properties ofthe industrial processes and of the armament systems became to be comparablewith similar properties of the natural processes. It concerns even such energy-consuming processes, as the natural cataclysms. On the one hand, this fact appearsastheevidenceofthegeneralprogressinthedevelopmentofhumanity.Ontheotherhand, this peculiarity evokes a serious misgiving, since it is threatening just thepossibility of the further existence of a human civilization. And such a misgivingis connected not only with a possibility of the global thermonuclear wars withunpredictable consequences, but also with the everyday activity on the security ofa public life. For example, one of the important factors is the hypothetical globalstate of climate of Earth. It is assumed that in the result of the large quantities ofsmoke and soot, which are carried out into the stratosphere through the spaciousfiresbytheexplosionof30–40%ofnuclearwar-charges,accumulatedintheworld,the temperature throughout the whole planet will be lowered down to the Arcticvalues, as a result of the essential increase of the quantity of reflected solar rays.The possibility of appearance of a nuclear winter was forecasted by Charles Saganin the USA and confirmed in Russia by the computations of V.V.Alexandrov.Theeverydayactivityonthesecurityofapubliclifeleadsbothtotheaccidentallarge-scale ecological catastrophes, and to the gradually accumulating pollutionof the environment (V.P. Dymnikov). Considered in the present monograph aresome fundamental problems connected with these subjects. Presented here arethe statistical methods of mathematical modeling for various models of the flowof fluid and gas, within the wide range of the characteristical parameters. Themodels of flow are ranging from the hypersonic flows of strongly rarefied gases(gaseous flows near the Earth’s satellites at the orbits and near the apparatusesdescendingfromtheorbit),whichareinfluencingtheecologicalstateofthenearestv
viMonte Carlo Methods in Mechanics of Fluid and Gasspace, and up to the turbulent flows modeling both the atmospheric phenomenaand the processes of flow about the modern flying machines. Described are themodern effective numerical methods, developed both by the authors themselvesand by other specialists and intended for the computer realization of these models.The problems considered belong to the classes of three-dimensional evolutionalproblems, based on the equations of mathematical physics, for the overwhelmingmajority of which are not proved even such a fundamental mathematical motions,as the theorems of existence and uniquity, even in the considerably simplifiedformulations. The study of such problems, at the present stage of the developmentof science, when the traditional analytical methods of investigation have, in acertain degree, exhausted themselves, is carried out, mainly, with the help of acomputational experiment.The revelation of the methods of statistical modeling (Monte Carlo) in variousareas of the applied mathematics is connected, as a rule, with the necessity ofsolutionofthequalitativelynewproblems,arisingfromtheneedsofpractice.Sucha situation appeared by the creation of the atomic weapon, at the initial stage of amastering of space, by the investigation of the phenomena of atmospheric optics,of the physical chemistry, and of the modeling of turbulence (G. von Neumann,Metropolis N., Unlam S., Vladimirov V.S., Sobol I.M., MArchuk G.I., ErmakovS.M., Mikhailov G.A., Bird G.A., Haviland J.K., Lavin M.D., Pullin D.I., KoganM.N., Perepukhov V.A., Beloserkovskii O.M., Yanitskii V.E., Ivanov M.S., andEropheevA.I.).As one of the more or less successful definitions of the Monte Carlo methods,it would be possible to present the following one:The Monte Carlo methods present in themselves the numerical methods ofsolutionofthemathematicalproblems(setsofthealgebraic,differential,orintegralequations) and the direct statistical modeling of the processes (physical, chemical,biological, economical, and social) with the help of obtainment (generation) of theaccidental numbers and the transformation of those.The book contains, in the reasonable proportions, those formulations and solu-tions, which already proved to be classical ones, as well as the results which haveendured the time control and were somewhat extended and supplemented in thelight of the last achievements in the corresponding areas of science. And, finally,this book fills in, by quite a natural way, the peculiar gap in the structure of com-putational aerodynamics, connected with a statistical modeling.The book was carried out within the frame of a scientific project “POISK”(“Search”), elaborated at the Department ofAeromechanics and Flying Techniqueof the Moscow Physico-Technical Institute (MPTI). The essence of this projectconsistsinthefollowing.Allaroundtheworldthetremendousnumberofresearches
Prefaceviiis working on the solution of fundamental and applied problems, connected withturbulence, especially with nonuniform and anisotropic one. Accumulated is thetremendous volume of factual material, and as rather actual point became that ofpreparation of a guide-book for orientation in that boundless sea of the theoretical,experimental,andnumericalresults.Attheabove-mentionedDepartmentofMPTIwas developed the project for such a guide-book and accompanying materials.Theproject’sstructurepresentsinitselfthecreationofbooks,containingtheanalysisofexperimental results, of the theoretical and computer-based methods. This projectis already partly realized.In particular, published is the book surveying the contemporary experimentalresearch on the dynamical structures within the turbulent boundary layer:(1) Yu.I. Khlopkov,V.A. Zharov, S.L. Gorelov, Coherent Structures in the Turbu-lent Boundary Layer. M., MPTI, 2002.Presented in this book, containing over 400 references, are the principles ofphysics of the dynamical processes in turbulent boundary layer, such as thephenomenonofbursting,theformationofstreaks,andtheprocessesoftransferof momentum and energy from the outer boundary of layer to that of the flowitself. Moreover, presented is the critical analysis of the foreign experimen-tal works, formulated are the actual problems. As it was found, the analysisof experimental investigations, conducted during a prolonged period (over40 years), revealed those essential features of the flows of fluid and gas, whichmightbeusedbytheconstructionofageneraltheoryoftheprocessesinvolved.The theoretical studies of turbulent flows are carried out during a long time,too. The considerable part of that time was devoted to the search of the mosteffective methods of problem’s solution. In the survey book,(2) Yu.I. Khlopkov,V.A. Zharov, S.L. Gorelov, Lectures on the Theoretical Meth-ods of Study of Turbulence. M., MPTI, 2005, are summed the results of thesestudies, and presented is the criticism of various methods, used at the earlierstage of the development of a theory. Thus, the reader is permitted to orientatehimself in the contemporary directions of study.The publishing house of MPTI has published also the survey book,(3) Yu.I. Khlopkov, V.A. Zharov, S.L. Gorelov, Renormgroup Methods of theDescription of Turbulent Motions of Incompressible Fluid. M., MPTI, 2006.Presented in this book is the survey of results of elaboration and application ofthe number of methods, named as renormgroup methods, for the constructionof models of turbulent flows of the incompressible fluid, both in the uniformand isotropic case, and in the case of a strong anisotropy and nonuniformity.Thebookisbasedonthestudyingofabout1000oftheoriginalworks,selected
viiiMonte Carlo Methods in Mechanics of Fluid and Gasfrom the totality of which were the most actual ones, according to the authorsopinion.Thelargestpartofcontentsisdevotedtothethreesub-networkmodelsof turbulence, which are widely used in the contemporary practical activityof various specialists in aerodynamics. The book is published as a textbookfor students, though it demands the considerable efforts for its understandingand is intended, actually, for the professors and postgraduates. At the presentmoment is prepared for publication “The Lecture Course on the Theory ofTurbulence”, which was presented at the Department of Aeromechanics andFlying Technique by the ProfessorV.N. Zhigulev and is devoted to the studieson that problem on the kinetical level. Further on, it is planned to carry out thesurvey and analysis of modern numerical methods, used by the modeling ofcomplicated unsteady flows of fluid and gas. The authors are expressing theirdeepest gratitude to the Russian Foundation of Fundamental Research, whichis supporting this project, especially useful for the young generation.The authors thank their colleagues M.N. Kogan, V.A. Zharov, S.L. Gorelov,I.V.Voronich, I.I. Lipatov, K.Yu. Gusarov, G.A.Tirskii, andV.V. Zasypalov for theparticipation in useful discussions and the observations spoken out, as well as thepostgraduates Olga Rovenskaja, Andrei Bukin, Tatjana Stanko, Anton Khlopkov,ZeiYahr,TunTun, and Ignat Ikrjanov for their help in our work. Our thanks also toMarina Spirkina andValentina Druzhinina for their help in putting the manuscriptinto shape.TheirspecialgratitudeauthorsexpresstoDrVsevolodPavlovichShidlovskyforhisqualifiedlaboronthetranslationofthisbook,andtotheeditorsoftranslation—Natalja Nosova and Irina Tarkhanova.
ContentsPrefacev0. Introduction11. The Main Equations andApproaches to Solutions of the Problemsin Rarefied Gas Dynamics231.1.1.2.The Main Equations in Rarefied Gas Dynamics . . . . . . . . . .The MainApproaches to the Construction of StatisticalAlgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Connection of the Stationary Modeling with the Solutionof Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Construction of the Method of Direct Statistical Modeling . . . .23251.3.26281.4.2. Development of the Numerical Methods of Solution of the LinearKinetic Equations302.1.2.2.The Perfection ofVGK Method (Vlasov, Gorelov, Kogan) . . . .Modification of the Vlasov’s Method for the Solutionof Linear Problems. . . . . . . . . . . . . . . . . . . . . . . .Method of Solution of the Linearized Boltzmann’s Equation . . .3035382.3.3. Methods of Solution of the Nonlinear Problems in RarefiedGas Dynamics433.1.Method of Solution of the Model Equation Basedon a Stationary Modeling . . . . . . . . . . . . . . . . . . . . .The Possibilities of the Scheme of Splitting for the Solutionof Kinetic Equations . . . . . . . . . . . . . . . . . . . . . . . .Increase of the Method’s Rate of Convergence . . . . . . . . . .Method by Belotserkovskii andYanitskii . . . . . . . . . . . . .433.2.4652543.3.3.4.ix
xMonte Carlo Methods in Mechanics of Fluid and Gas4. Modeling of the Flow of Continuous Media584.1.Procedure of the Monte Carlo Methods for Modelingthe Flows of Rarefied Gas and Continuous Medium . . . . . . .Method “Relaxation–Transfer” for a Solution of the Problemsof Gas Dynamics in the Wide Range of the Degreeof Rarefaction of a Medium (see Kogan et al.83) . . . . . . . . .Modeling of the Flows of Nonviscous Perfect Gas . . . . . . . .584.2.62664.3.5. Solution of the Navier–Stokes Equations (Petrov133−139)725.1.Formulation of the Problem, Initial and Boundary Conditionsfor the Navier–Stokes Equations in the Form by Helmholtz . . .The General Properties of the Vertical FlowArisingby the Instantaneous Start of a Body from theState of Rest . . . . . . . . . . . . . . . . . . . . . . . . . . . .Initial Conditions for the Problem of the Instantaneous Startof a Body in a Viscous Fluid . . . . . . . . . . . . . . . . . . . .The GeneralAlgorithm of the Numerical Solutionof an Initial–Boundary Problem for the Navier–StokesEquations in the form by Helmholtz . . . . . . . . . . . . . . . .Solution of the Cauchy Problem for the Fokker–PlankEquation at Small Interval of Time . . . . . . . . . . . . . . . .The Numerical Solution of the Fokker–PlankEquation by the Method of Direct Statistical Modeling . . . . . .725.2.745.3.785.4.805.5.885.6.956. Studies of the Weakly Perturbed Flows of Rarefied Gas1036.1.6.2.Determination of the Velocity of Slip . . . . . . . . . . . . . . . 103Solution of the Problem of the Feeble Evaporation(Condensation) from the Plane Surface(see Korovkin, Khlopkov104) . . . . . . . . . . . . . . . . . . . 106The Slow Motion of a Sphere in Rarefied Gas(Brownian Motion). . . . . . . . . . . . . . . . . . . . . . . . 108The Coefficient of Diffusion and the Mean Shiftingof a Brownian Particle in the Rarefied Gas (see Khlopkov106) . . 1106.3.6.4.7. Study of the FlowsAbout Different Bodies in Transitional Regime1147.1.7.2.FlowsAbout the Planar Bodies . . . . . . . . . . . . . . . . . . 115FlowsAboutAxisymmetrical Bodies . . . . . . . . . . . . . . . 119
Contentsxi7.3.Influence of the Evaporation (Condensation) on theAerodynamical Resistance of a Sphere by the SupersonicFlowAbout It . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Computation of the Steady Regime of a FlowAbout a Bodyand of the Profile Resistance in a Viscous Gas(SeeA.S. Petrov) . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.4.8. Determination of theAerodynamical Characteristics of the ReturnableSpace Systems (RSS)1388.1.8.2.Methodics of the Description of a SurfaceMethodics of Calculation of theAerodynamicalCharacteristics of the FlyingApparatus in the Conditionsof a Free-Molecular Flow . . . . . . . . . . . . . . . . . . . . . 142The Engineering Methodics of the ComputationofAerodynamical Characteristics of the Bodiesof Complicated Form in a Transitional Regime(see Galkin, Eropheev, Tolstykh85) . . . . . . . . . . . . . . . . 143The Results of the FlowAbout a Hypersonic FlyingApparatus “Clipper” (see Voronich, ZeyYar225). . . . . . . . . . . . 1388.3.8.4.. . . . . . . . . 1459. The FlowAbout Blunted Bodies with theAddition of Heat(see Vorovich, Moiseev)1659.1.9.2.9.3.9.4.9.5.The Main Features of a Method . . . . . . . . . . . . . . . . . . 165Description of theAlgorithm . . . . . . . . . . . . . . . . . . . 167TheApproximational Properties . . . . . . . . . . . . . . . . . . 170TheAlgorithm and the Nets . . . . . . . . . . . . . . . . . . . . 172Direct Statistical Modeling of the Inviscid FlowsAbout Blunted Bodies by the Presenceof EnergyAddition. . . . . . . . . . . . . . . . . . . . . . . . 17510. The General Models of Description of the Turbulent Flows18710.1. Theoretical Methods of the Description of Turbulence . . . . . . 18710.2. Coherent Structures in the Turbulent Boundary Layer(see Khlopkov, Zharov, Gorelov205) . . . . . . . . . . . . . . . . 19410.3. The Description of Turbulence with the Help of a Modelof the Three-Wave Resonance . . . . . . . . . . . . . . . . . . . 20410.4. The Fluidical Model of the Description of Turbulence(Belotserkovskii,Yanitskii) . . . . . . . . . . . . . . . . . . . . 208
xiiMonte Carlo Methods in Mechanics of Fluid and Gas11. Studies of the Turbulent Flow of Fluid and Gas21111.1. Modeling of a Turbulent Transition within the BoundaryLayer Using Monte Carlo Method (see Zharov, Tun Tun,Khlopkov223). . . . . . . . . . . . . . . . . . . . . . . . . . . 21111.2. Study of the Dissipation ofTurbulent Spots (see Belotserkovskii,Yanitskii, Bukin12,221) . . . . . . . . . . . . . . . . . . . . . . . 21811.3. Evolution of theVertical System in the Rarefied Gas (see Roven-skaya, Voronich, Zharov222) . . . . . . . . . . . . . . . . . . . . 21912. The Possible Directions of Development of the Methodsof Statistical Study22812.1. Development of the Methods of Solutionof Linear Problems12.2. Use of the Possibilities of the Model Equations . . . . . . . . . . 23212.3. Modeling of the Flows of Continuous Medium . . . . . . . . . . 23512.4. Modeling of the Turbulent Flows of Fluid and Gas . . . . . . . . 24012.5. Parallelization of the StatisticalAlgorithms (Bukin,Voronich, Shtarkin) . . . . . . . . . . . . . . . . . . . . . . . . 245. . . . . . . . . . . . . . . . . . . . . . . . 228Conclusions253References257
0. Introduction0.1The General Scheme of Monte Carlo MethodsThe first publication on the use of Monte Carlo methods was made by Hall1in1873 while organizing the stochastic process of experimental determination of πnumber by means of throwing of needle on the sheet of lined paper. The brightexample of the application of Monte Carlo methods consists in the use of ideaby J. von Neumann realized in 1940s of the past century, by the modeling of theneutron’strajectoriesinLosAlamoslaboratory.InspiteofthefactthatMonteCarlomethodsareconnectedwithalargeamountofcomputations,thelackofelectronicalcomputational technique did not embarrass investigators in neither case duringtheir application of the methods of statistical modeling, because in both cases, thestudy was concerned with realization of accidental processes. And these methodsacquired their romantic title by the name of enclave of Monaco, which is famousduetoitsgamblinghouses,wheretheprincipalobjectisroulette—themostperfectinstrument for obtaining the accidental numbers. And the first publication with asystematic exposition of that matter was made in 1949 by Metropolis and Ulam,2where Monte Carlo method was applied to a solution of linear integral equations.And in that publication was implicitly revealed the problem of the passing ofneutrons through the substance. When speaking of Russia, the publications onMonte Carlo methods began to appear actively after the International Conferencein Geneva devoted to the peaceful applications of nuclear energy. As one of thefirst one could mention the work by Vladimirov and Sobol.3Beginning with theearliest1970s,sidebysidewiththeregularmethodsthoseofMonteCarloobtainedtheir proper place in computational mathematics (Marchuk, Samarskii, Popov,Belotserkovskii, Bakhvalov, Ermakov, Mikhailov, Sobol, Bird, Haviland, Kogan,Perepukhov, and Janitskii).The general scheme of Monte Carlo method is based on the central limitingtheorem of the theory of probability, which states that the accidental quantityY =?ni=1XiwhichisequaltoasumofthelargeamountN ofarbitraryaccidentalnumbers Xihaving identical mathematical expectations m and dispersions σ2, is1
2Monte Carlo Methods in Mechanics of Fluid and Gasalways distributed according to a normal law with mathematical expectation N ·mand dispersion N · σ2.Let us assume that we wish to find a solution of some equation or a result ofsome process I. If the accidental quantity ξ with a probability density p is built upin such a way that the mathematical expectation for this quantity would be equalto the solution we are looking for, M(ξ) = I, then that would give the simple wayof estimation of both the solution and the error,I = M(ξ) ≈1NN?i=1ξi±3σ√N.Following from above are the general properties of the methods:— the absolute convergence to a solution as 1/N;— the strong dependence of the error ε on the number of trials, as ε ≈is, for the diminishment of the error on one order it is necessary to increase thenumber of trials on two orders);— the main way of the error’s diminishment consists in the maximal diminish-ment of dispersion or, in the other words, it is necessary to draw as nearas possible the probability density p(x) of the accidental quantity ξ to themathematical formulation of the problem or to the physics of a phenomenonmodeled;— the error does not react on the problem’s dimensionality (by the use of finite-difference methods the transition from the one-dimensional problem to thethree-dimensional one the number of computations would be increased on twoorders, while in Monte Carlo methods the number of computations remains onthe same order);— the simple structure of computational algorithm (number N of the single-typecomputations by the realization of accidental quantity);— moreover, the construction of accidental quantity ξ might be based on thephysical nature of process only, and would not demand, as it is in the regularmethods, the compulsory formulation of the equation; such a quality becomesmore and more actual for modern problems.1√N(thatThemainpropertiesofMonteCarlomethods,aswellastheconditionsatwhichthey yield or surpass the traditional finite-difference approaches, might be demon-strated by the application to some simple problem, for example, to the problem ofthe computation of an integral?bI =af(x)dx,
Introduction3where x, a, and b are vectors in a n-dimensional Euclidian space. Let us build upthe accidental quantity ξ with density p(x) in such a way that the mathematicalexpectation?∞would occur to be equal to our integral I. Then, if within the proper limits onewould choose ξ = f(x)/p(x), then the central limiting theorem would giveM(ξ) =−∞ξ · p(x)dx,I =1NN?i=1ξi±3ε√N.Thus, we have as the first: The computation of the integral I might be interpreted,from one side, as a solution of mathematically formulated problem, and, from theother side, as a direct modeling of determination of a volume found under thefunction f(x).The second: The computation of the one-dimensional integral I1by the MonteCarlomethodcorrespondstotheintegral’scomputationbythemethodofrectangleswith a step ?x ≈ 1/N and an error O(?x). In principle, by the sufficiently goodfunction f(x) in one-dimensional case the integral I1might be calculated withaccuracy O(?x2) with trapezoids, with accuracy O(?x3) with parabola’s, and,generally speaking, with any predicted accuracy. In multi-dimensional case, thedifficulties of use of the high-order schemes become to acquire such an essentialcharacter that by the computation of n-dimensional integrals Inwith n ≥ 3 thehigh-order schemes are used just very rarely.Let us build up the correspondence in effectiveness between regular methodsandstatisticalones.Letitbethatnistheproblem’sdimensionality,Y —thenumberof knots at the axis, R = Yn— the total number of knots for regular methods,q — the order of accuracy of the scheme, N — the number of statistical trials, ν —the number of operations for a treatment of a single knot, εL= Y−q— the errorof computations for regular methods, εK= N−1/2— the error of computationsfor Monte Carlo, L(ε) = ν · R = ν · ε−n/q— the number of operations bythe problem’s solution with regular methods, K(ε) = ν · N = ν · ε−2— thenumber of operations by the use of Monte Carlo method. For the case of one andthe same number of operations by the computation of solution by one or anothermethod one would obtain the relation n = 2q. This means that with n ≥ 3, whenmainly the first-order schemes are used, the Monte Carlo methods occur to bepreferable.
4Monte Carlo Methods in Mechanics of Fluid and Gas0.2Special Position of Monte Carlo Methods in ComputationalAerodynamicsDynamics of the rarefied gases is treated by means of well-known integro-differential equation — Boltzmann equation:?where f = f(t,x,y,z,ξx,ξy,ξz) is the function of molecule’s distribution inrespect of time, coordinates, and velocities, f?,f?sponding to the molecule’s velocities after collision, ξ?,ξ?of molecules by collisions in pairs, ? g =?ξ −?ξ1=?ξ?−?ξ?tanceandazimuthalanglebythecollisionsofparticles.Thecomplicatednonlinearstructure of the integral of collisions and large number of variables (in the generalcase — 7) do create the essential difficulties for the analysis, including the numer-ical one, and, practically, lead to the exclusion of the finite-differential approachfrom the process of the solution of serious problems.At the same time, the multi-dimensionality and probabilistical nature of kinetic processes create the naturalground for the application of Monte Carlo methods.Historically, the application of Monte Carlo methods to the computationalaerodynamics was initiated in TSAGI by the pioneering works by M.N. KoganandV.A. Perepukhov devoted to the modeling of free-molecular flows about spaceobjects, in the part of trajectory of their orbital flight. Such a modeling is just thesimplest form of rarefied gas dynamics. The further development of the statisticalcomputational methods was realized in the following three directions:∂f∂t+?ξ∇f =(f?· f?1− f · f1) · ? g · b · db · dε · d?ξ1,(0.2.1)1— distribution functions corre-1, ? g — relative velocities1, b, and ε — aiming dis-Professor M.N. Kogan. 1965.
Introduction5— use of the Monte Carlo methods for the calculation of collision integrals foundin the regular finite-difference schemes designed for the solution of kineticequations;— direct statistical modeling of the physical phenomenon which is splitted intwo approaches: modeling of the trajectories of “trial particles” according toHaviland4and modeling of the evolution of “ensemble of particles” accordingto Bird5;— construction of the accidental process of the type of a procedure by Ulamand Neumann described in Ref. 6 and corresponding to a solution either oflinearized kinetic equation,8or of Master Equation by Kac.7The probabilistical nature of the aerodynamics of rarefied gases, which is soimportant for application and development of the numerical schemes of MonteCarlo, follows quite naturally from the general principles of kinetic theory andstatistical physics.Laureate of State prizeV.A. Perepukhov. 1965The reasoning cited below might be, quite perfectly, looked at as the levels ofcompleteness of description of a large molecular system. Further on, these levelswill be needed for a construction of the effective methods of statistical modeling.The most detailed level of description is presented by a dynamical system. Todescribe such a system which comprises of a large number of elements N (notethatmoleculargasisjustsuchasystemwithN ≈ 1023molecules),itisnecessaryto
6Monte Carlo Methods in Mechanics of Fluid and GasFig. 0.1Evolution of the dynamical system in 6N-dimensional space.settheinitialcoordinatesandvelocitiesofeachmolecule(? rj,? vj)andtheequationsof evolution of this system:md2? rjdt2=N?i?=jRij.(0.2.2)Solution of such a system appears to be quite unreal problem, even for a stronglyrarefied gas — at the height of 400–600km (the most popular orbits of satellites)one cubic centimeter contains 109molecules. For this reason one comes to the lesscomplete,thatis,statisticaldescriptionofthebehaviorofthesystem.Inaccordanceto a Gibbs formalism one considers not a single system, but the ensemble of themin 6N-dimensional ?-space (Fig. 0.1), with system’s distributed according to theN-particle distribution function F(t,? r1,...,? rN,? ν1,...,? νN) = FN, of which thesense is that of a probability for a system to be in the time moment t at the point? r1,? r2,...,? rN,? ν1,? ν2,...,? νN, in vicinity of d? r1,...,d? rNd? ν,...,d? νNwe havedW = FNd? r1,...,d? rNd? ν1,...,d? νN.Such an ensemble is described by the famous Liouville equation:∂FN∂t+N?i=1νi∂FN∂ri+N?i?=jN?i=1Rij∂FNm∂νi= 0.(0.2.3)And beginning with that moment the Liouville equation and all other kinetic equa-tions following from the Bogoljubov’s chain, including the last link — Boltzmannequation—possesstheprobabilisticalnature.AndinspiteofthefactthatEq.(0.2.3)is simpler than system (0.2.2), it takes into consideration the N-particle collisionsofmoleculesandalsoremainstobeextremelycomplicatedforapracticalanalysis.The transition to a less detailed level of description is connected with the furthercoarsening of system’s description with the help of s-particle distribution func-tions Fs =?FNd? rs+1···d? rNd? νs+1···d? νN, which determine the probability of
Introduction7the simultaneous revelation of s particles independently of the state of the remain-ing N − s particles. Following the ideas of Bogoljubov one obtains the chain ofinterconnected equations:∂Fs∂t+s?i=1νi∂Fs∂ri+s?0=1s?j?=iRij∂Fsm∂νi= −s?i=1(N−s)∂∂νi?Ri,s+1mFs+1drs+1dνs+1,(0.2.4)uptotheone-particledistributionfunctionF1= f(t,? r,?ξ)fortheBoltzmann’sgas,taking into account only by-pair collisions:∂f∂t+?ξ∂f∂? r+R12∂fm∂?ξ= −∂∂?ξ?R12mF2d? r1d?ξ1.Following Boltzmann we shall consider the molecules to be spherically sym-metrical and, adopting the hypothesis of a molecular chaos, F2(t,? r,? ν1,? ν) =F1(t,? r,? ν1)F1(t,? r,? ν2), one obtains Eq. (0.2.1).AsratherinterestingonemightconsidertheparticularcaseofLiouville’sequa-tion (0.2.3) and Bogoljubov’s chain (0.2.4) for the spatially uniform gas consistingof the limited number of particles.At the terminal link this case leads to obtainingthe famous equation by Kac — “Master Equation”7:?whereφ1andφ2areone-andtwo-particledistributionfunctions.UnliketheBoltz-mann’s equation, Eq. (0.2.5) is linear, and this fact will be used by the constructionand estimation of the effective computational schemes of direct statistical model-ing. When coming back to the Boltzmann equation, from the determination of thefunction f it would be easy to obtain all the macroscopic parameters. Thus, thenumber n of molecules within the unit volume of gas is equal to?Similarly to that, the mean velocity of molecules, stress tensor and vector of theflow of energy are defined by the relations?Pij= m?∂φ1(t,?ξ1)∂t=N − 1N[φ2(t,?ξ?1,?ξ?2) = φ2(t,?ξ1,?ξ2)] · g12dσ12d?ξ,(0.2.5)n(t,x) =f(t,x,ξ)dξ.u(t,x) = (1/n)ξf(t,x,ξ)dξ,?cicjf(t,x,ξ)dξ,qi= (m/2)c2cif(t,x,ξ)dξ,
8Monte Carlo Methods in Mechanics of Fluid and Gaswhere c = ξ−u is thermal velocity of molecules.The mean energy of the thermalmotion of molecules is usually characterized by temperature?By way of application to the Boltzmann equation procedure of Enskog and Chap-man one obtains the hydrodynamical level of description. Thus, on that level thedescription corresponds to Navier–Stokes equations:32kT =1nmc22f(t,x,ξ)dξ.∂ρ∂e+∂ρui?∂∂xiui=1?= 0,?∂?∂Pij= pij+ ρijp,?∂ui∂t+ uj∂∂xjρ∂Pij∂xj+Xm,32Rρ∂t+ uj∂xjT = −∂qj∂xj− Pij∂uj∂xj,p = ρRT,−2(0.2.6)pij= µ∂xj+∂uj∂xi3δij∂ur∂xr?,qi= −λ∂T∂xiand Euler’s equations:∂ρ∂t+∂ρui?∂t+ uj∂xiui= −1?p = ρRT.= 0,?∂32Rρ∂t+ uj?∂∂∂xjρgrad · p,(0.2.7)∂∂xjT = −p · divu,Followingthegenerallogicofthepresentexpositiononemightassumethatdynam-ics of continuum, as a particular case of kinetical approach to the treatment ofmotion of a gas, possesses the features of statistical nature and permits the realiza-tion of statistical modeling, just what will be demonstrated below.0.3The Position of Monte Carlo Methods in Modern MathematicsThe singular features of Monte Carlo methods cited in the preceding sectionjust lead to the necessity of marking the position of methods of the statis-tical modeling in modern mathematics. Undoubtedly, at the present moment
Introduction9the priority is firmly kept by the traditional theoretical approaches (seeworks by Sadovnichij,258,259Matrosov,270Zhuravlev, Fljorov,285and others)and finite-differential approaches (see Marchuk,236Bakhvalov,235,247Kholodov,Magomedov...