a hybrid method for hydrodynamic kinetic flow part ii coupling of hydrodynamic and kinetic models in this work we present a non stationary domain decomposition algorithm for multiscale hydrodynamic kinetic problems in which the knudsen number may span from equilibrium to highly rarefied regimes our approach is characterized by using the full boltzmann equation for the kinetic regime the compressible euler equations for equilibrium with a buffer zone in which the bgk es equation is used to represent the transition between fully kinetic to equilibrium flows in this fashion the boltzmann solver is used only when the collision integral is non stiff and the mean free path is of the same order as the mesh size needed to capture variations in macroscopic quantities thus in principle the same mesh size and time steps can be used in the whole computation moreover the time step is limited only by convective terms since the boltzmann solver is applied only in wholly kinetic regimes we use the reduced noise dsmc scheme we have proposed in part i of the present work this ensures a smooth exchange of information across the different domains with a natural way to construct interface numerical fluxes several tests comparing our hybrid scheme with full boltzmann dsmc computations show the good agreement between the two solutions on a wide range of knudsen numbers

ricci collineations of static spherically symmetric spacetimes the ricci collineations of static spherically symmetric spacetimes are classified and their relationship with isometries is discussed a general theorem about this relationship is stated and its extension to all spherically symmetric spacetimes is discussed

the bgk model with external confining potential existence long time behaviour and time periodic maxwellian equilibria we study global existence and long time behaviour for the inhomogeneous nonlinear bgk model for the boltzmann equation with an external confining potential for an initial datum f0 0 with bounded mass entropy and total energy we prove existence and strong convergence in l1 to a maxwellian equilibrium state by compactness arguments and multipliers techniques of particular interest is the case with an isotropic harmonic potential in which boltzmann himself found infinitely many time periodic maxwellian steady states this behaviour is shared with the boltzmann equation and other kinetic models for all these systems we study the multistability of the time periodic maxwellians and provide necessary conditions on f0 to identify the equilibrium state both in l1 and in lyapunov sense under further assumptions on f these conditions become also sufficient for the identification of the equilibrium in l1

stationary solutions for the ellipsoidal bgk model in a slab abstract we address the boundary value problem for the ellipsoidal bgk model of the boltzmann equation posed in a bounded interval the existence of a unique mild solution is established under the assumption that the inflow boundary data does not concentrate too much around the zero velocity and the gas is sufficiently rarefied

symmetries of the energy momentum tensor of spherically symmetric lorentzian manifolds matter collineations of spherically symmetric lorentzian manifolds are considered these are investigated when the energy momentum tensor is nondegenerate and also when it is degenerate we have classified space times admitting higher symmetries and space times admitting so 3 as the maximal isometry group for the nondegenerate case we obtain either four six seven or ten independent matter collineations in which four are isometries and the rest are proper the results of the previous paper sharif and sehar gen relativ gravit 35 1091 2003 are recovered as a special case it is worth noting that we have also obtained two cases where the energy momentum tensor is degenerate but the group of matter collineations is finite dimensional i e four or ten

ellipsoidal bgk model for polyatomic molecules near maxwellians a dichotomy in the dissipation estimate abstract we consider the global existence and asymptotic behavior of classical solutions to the ellipsoidal bgk model for polyatomic molecules when the initial data starts sufficiently close to a global polyatomic maxwellian we observe that the linearized relaxation operator is decomposed into a truly polyatomic part and an essentially monatomic part leading to a dichotomy in the dissipative property in the sense that the degeneracy of the dissipation shows an abrupt jump as the relaxation parameter reaches zero accordingly we employ two different sets of micro macro system to derive the full coercivity and close the energy estimate

killing vectors of static spherically symmetric metrics an error in a previous theorem j math phys 28 1019 1987 29 525 1988 is corrected and the theorem is extended

symmetries of the energy momentum tensor of cylindrically symmetric static space times we investigate matter symmetries of cylindrically symmetric static space times these are classified for both cases when the energy momentum tensor is nondegenerate and also when it is degenerate it is found that the nondegenerate energy momentum tensor gives either three four five six seven or ten independent matter collineations in which three are isometries and the rest are proper the worth mentioning cases are those where we obtain the group of matter collineations finite dimensional even the energy momentum tensor is degenerate these are either three four five or ten some examples are constructed satisfying the constraints on the energy momentum tensor

numerical comparison between the boltzmann and es bgk models for rarefied gases abstract rarefied gas flows obey the boltzmann equation but numerical simulations of this equation are not always possible so that simpler models have been introduced the es bgk equation is one of these models it gives the correct transport coefficients for the navier stokes approximation so that boltzmann or es bgk simulations are expected to give the same results for dense gases but in the case of a rarefied flow complete numerical comparisons are needed in this paper we present numerical comparisons between the two models in transitional regimes where the es bgk model is expected to be useful for reentry flows around a compression ramp and a plate we also emphasize that the es bgk model gives flow predictions closer to the boltzmann result than the simpler bgk model

optimal exponential decay for the linearized ellipsoidal bgk model in weighted sobolev spaces this paper deals with the asymptotic behavior of solution to the linearized ellipsoidal bgk model in torus we prove that the solution converges exponentially to the equilibrium in the weighted sobolev spaces with polynomial weight our exponential decay rate e lambda t is optimal in the sense that lambda 0 equals to the spectral gap of the linearized operator in the standard hilbert space our strategy is taking advantage of the quantitative spectral gap estimates in a smaller reference hilbert space the factorization method and the enlargement of the functional space for the associated semigroup

an implicit monte carlo method for rarefied gas dynamics for the space homogeneous boltzmann equation we formulate a hybrid monte carlo method that is robust in the fluid dynamic limit this method is based on an analytic representation of the solution over a single time step and involves implicit time differencing derived from a suitable power series expansion of the solution a generalized wild expansion a class of implicit yet explicitly implementable numerical schemes is obtained by substituting a maxwellian distribution in place of the high order terms in the expansion the numerical solution is represented as a convex combination of a non equilibrium particle distribution and a maxwellian the hybrid distribution is then evolved by monte carlo using the implicit formulation for the time evolution computational simulations of spatially homogeneous problems by our method are presented here for the kac model and for the variable hard sphere model including maxwell molecules comparison to exact solutions and to direct simulation monte carlo dsmc computations shows the robustness and the efficiency of the new method

microscopically implicit macroscopically explicit schemes for the bgk equation in this work a new class of numerical methods for the bgk model of kinetic equations is introduced the schemes proposed are implicit with respect to the distribution function while the macroscopic moments are evolved explicitly in this fashion the stability condition on the time step coincides with a macroscopic cfl evaluated using estimated values for the macroscopic velocity and sound speed thus the stability restriction does not depend on the relaxation time and it does not depend on the microscopic velocity of energetic particles either with the technique proposed here the updating of the distribution function requires the solution of a linear system of equations even though the bgk model is highly non linear thus the proposed schemes are particularly effective for high or moderate mach numbers where the macroscopic cfl condition is comparable to accuracy requirements we show results for schemes of order 1 and 2 and the generalization to higher order is sketched

diffusion approximation and computation of the critical size this paper is devoted to the mathematical definition of the extrapolation length which appears in the diffusion approximation to obtain this result we describe the spectral properties of the transport equation and we show how the diffusion approximation is related to the computation of the critical size the paper also contains some simple numerical examples and some new results for the milne problem introduction the computation of the critical size and the diffusion approximation for the transport equation have been closely related and this is due to the following facts first the computation of the critical size is much easier for the diffusion approximation than for the original transport equation second for the critical size one can consider a host media x nxq with x0 given and 17 a positive number the problem of the critical size is then reduced to the computation of the parameter zj it turns out that when the transport operator is almost conservative the critical value of the parameter 17 is large and it is exactly for this range of value that the diffusion approximation is accurate on the other hand the physical boundary condition for the diffusion approxi mation is of the form

cauchy problem for the ellipsoidal bgk model of the boltzmann equation ellipsoidal bgk model es bgk is a generalized version of the original bgk model designed to reproduce the physically correct prandtl number in the navier stokes limit in this paper we study the cauchy problem for the es bgk model under the condition of finite initial mass energy and entropy equivalence type estimates for the temperature tensor are crucially used

fluid dynamic limits of kinetic equations ii convergence proofs for the boltzmann equation using relative entropy estimates about an absolute maxwellian it is shown that any properly scaled sequence of diperna lions renormalized solutions of some classical boltzmann equations has fluctuations that converge to an infinitesimal maxwellian with fluid variables that satisfy the incompressibility and boussinesq relations moreover if the initial fluctuations entropically converge to an infinitesimal maxwellian then the limiting fluid variables satisfy a version of the leray energy inequality if the sequence satisfies a local momentum conservation assumption the momentum densities globaly converge to a solution of the stokes equation a similar discrete time version of this result holds for the navier stokes limit with an additional mild weak compactness assumption the continuous time navier stokes limit is also discussed 01993 john wiley sons inc

on the cauchy problem of the vlasov poisson bgk system global existence of weak solutions global in time existence of weak solutions to the cauchy problem of the three dimensional vlasov poisson bgk system is shown for initial data belonging to the space lp 3 3 with p 9 and having finite second order velocity moments this result solves partially the well posed problem for the vlasov poisson bgk system proposed by b perthame higher moments for kinetic equations the vlasov poisson and fokker planck cases math meth appl sci 13 441 452 1990

classical solutions for the ellipsoidal bgk model with fixed collision frequency abstract we establish the existence of global in time smooth solutions for the ellipsoidal bgk model which is a variant of the bgk model for the boltzmann equation designed to yield the correct prandtl number in the hydrodynamic approximation at the navier stokes level for this we carefully design a function space which captures the growth of the solution in a weighted sobolev norm and show that the ellipsoidal relaxation operator is lipschitz continuous in the induced metric this approach is restricted to the case when the collision frequency does not depend on the macroscopic field but no smallness on the initial data is required

construction of bgk models with a family of kinetic entropies for a given system of conservation laws we introduce a general framework for kinetic bgk models we assume to be given a system of hyperbolic conservation laws with a family of lax entropies and we characterize the bgk models that lead to this system in the hydrodynamic limit and that are compatible with the whole family of entropies this is obtained by a new characterization of maxwellians as entropy minimizers that can take into account the simultaneous minimization problems corresponding to the family of entropies we deduce a general procedure to construct such bgk models and we show how classical examples enter the framework we apply our theory to isentropic gas dynamics and full gas dynamics and in both cases we obtain new bgk models satisfying all entropy inequalities

weighted l bounds and uniqueness for the boltzmann bgk model we prove rather general l bounds for hydrodynamical fields in terms of weighted l norms of the kinetic density we use these estimates to prove l bounds and uniqueness for the solution of the bgk equation which is a simple relaxation model introduced by bhatnagar gross krook to mimic boltzmann flows

ricci collineation vectors in fluid space times the properties of fluid space times that admit a ricci collineation vector rcv parallel to the fluid unit four velocity vector ua are briefly reviewed these properties are expressed in terms of the kinematic quantities of the timelike congruence generated by ua the cubic equation derived by oliver and davis ann inst henri poincare 30 339 1979 for the equation of state p p of a perfect fluid space time that admits an rcv which does not degenerate to a killing vector is solved for physically realistic fluids necessary and sufficient conditions for a fluid space time to admit a spacelike rcv parallel to a unit vector na orthogonal to ua are derived in terms of the expansion shear and rotation of the spacelike congruence generated by na perfect fluid space times are studied in detail and analogues of the results for timelike rcvs parallel to ua are obtained properties of imperfect fluid space times for which the energy flux vector qa vanishes and na is a spacelike eigenvector of the aniso

sector delayed hopf type mixed mode oscillations in a prototypical three time scale model we consider a three dimensional three time scale system that was first proposed by krupa et al 2008 under the additional assumption that two singular perturbation parameters are present in the equations while the presence of three scales was shown to give rise to canard induced periodic mixed mode oscillations mmos desroches et al 2012 in the parameter regime studied by krupa et al 2008 we additionally observe mixed mode patterns that display delayed hopf type behaviour ne shtadt 1987 we present analytical and numerical evidence for the occurrence of stable periodic dynamics that realises both mechanisms and we discuss the transition between them to the best of our knowledge the resulting mixed sector delayed hopf type mmo trajectories represent a novel class of mixed mode dynamics in singularly perturbed systems of ordinary differential equations

addendum symmetries of the energy momentum tensor in recent papers j math phys 44 5142 2003 45 1518 2003 45 1532 2004 we have discussed matter symmetries of nonstatic spherically symmetric space times static plane symmetric space times and cylindrically symmetric static space times these have been classified for both cases when the energy momentum tensor is nondegenerate and also when it is degenerate here we add up some consequences and the missing references about the ricci tensor

the classification of static plane symmetric space times according to their ricci collineations static plane symmetric space times are classified according to their ricci collineations rcs their relation with isometries of the space times is established unlike the spherically symmetric space times some metrics are found for a nondegenerate ricci tensor which have more rcs than isometries

non linear diffusions as limit of kinetic equations with relaxation collision kernels kinetic transport equations with a given confining potential and non linear relaxation type collision operators are considered general monotone energy dependent equilibrium distributions are allowed with a chemical potential ensuring mass conservation existence and uniqueness of solutions is proved for initial data bounded by equilibrium distributions the diffusive macroscopic limit is carried out using compensated compactness theory the results are drift diffusion equations with non linear diffusion the most notable examples are of the form partial t rho nabla cdot nabla rho m rho nabla v ranging from porous medium equations to fast diffusion with the exponent satisfying 0 m 5 3 in mathbb r 3

fractional diffusion limit for collisional kinetic equations this paper is devoted to diffusion limits of linear boltzmann equations when the equilibrium distribution function is a maxwellian distribution it is well known that for an appropriate time scale the small mean free path limit gives rise to a diffusion equation in this paper we consider situations in which the equilibrium distribution function is a heavy tailed distribution with infinite variance we then show that for an appropriate time scale the small mean free path limit gives rise to a fractional diffusion equation

local analysis near a folded saddle node singularity abstract folded saddle nodes occur generically in one parameter families of singularly perturbed systems with two slow variables we show that these folded singularities are the organizing centers for two main delay phenomena in singular perturbation problems canards and delayed hopf bifurcations we combine techniques from geometric singular perturbation theory the blow up technique and from delayed hopf bifurcation theory complex time path analysis to analyze the flow near such folded saddle nodes in particular we show the existence of canards as intersections of stable and unstable slow manifolds to derive these canard results we extend the singularly perturbed vector field into the complex domain and study it along elliptic paths this enables us to extend the invariant slow manifolds beyond points where normal hyperbolicity is lost furthermore we define a way in way out function describing the maximal delay expected for generic solutions passing through a folded saddle node singularity branch points associated with the change from a complex to a real eigenvalue structure in the variational equation along the critical slow manifold make our analysis significantly different from the classical delayed hopf bifurcation analysis where these eigenvalues are complex only

convergence of a semi lagrangian scheme for the bgk model of the boltzmann equation recently a new class of semi lagrangian methods for the bgk model of the boltzmann equation has been introduced f filbet and g russo kinet relat models 2 2009 pp 231 250 g russo and p santagati a new class of large time step methods for the bgk models of the boltzmann equation arxiv 1103 5247 p santagati high order semi lagrangian methods for the bgk model of the boltzmann equation ph d thesis university of catania italy 2007 these methods work in a satisfactory way either in a rarefied or a fluid regime moreover because of the semi lagrangian feature the stability property is not restricted by the cfl condition these aspects make them very attractive for practical applications in this paper we prove that the discrete solution of the scheme converges in a weighted l 1 norm to the unique smooth solution by deriving an explicit error estimate

convergence of a discrete velocity model for the boltzmann bgk equation abstract we prove the convergence of a conservative and entropic discrete velocity model for the bathnagar gross krook bgk equation in this model the approximation of the maxwellian is based on a discrete entropy minimization principle the main difficulty due to its implicit definition is to prove that this approximation is consistent we also demonstrate the existence and uniqueness of a solution to the discrete velocity model by using a fixed point theorem finally the model is written in a continuous equation form and we prove the convergence of its solution toward a solution of the bgk equation

a hybrid method for hydrodynamic kinetic flow part i a particle grid method for reducing stochastic noise in kinetic regimes in this work we present a hybrid particle grid monte carlo method for the boltzmann equation which is characterized by a significant reduction of the stochastic noise in the kinetic regime the hybrid method is based on a first order splitting in time to separate the transport from the relaxation step the transport step is solved by a deterministic scheme while a hybrid dsmc based method is used to solve the collision step such a hybrid scheme is based on splitting the solution in a collisional and a non collisional part at the beginning of the collision step and the dsmc method is used to solve the relaxation step for the collisional part of the solution only this is accomplished by sampling only the fraction of particles candidate for collisions from the collisional part of the solution performing collisions as in a standard dsmc method and then projecting the particles back onto a velocity grid to compute a piecewise constant reconstruction for the collisional part of the solution the latter is added to a piecewise constant reconstruction of the non collisional part of the solution which in fact remains unchanged during the relaxation step numerical results show that the stochastic noise is significantly reduced at large knudsen numbers with respect to the standard dsmc method indeed in this algorithm the particle scheme is applied only on the collisional part of the solution so only this fraction of the solution is affected by stochastic fluctuations but since the collisional part of the solution reduces as the knudsen number increases stochastic noise reduces as well at large knudsen numbers

relaxation oscillation and canard explosion abstract we give a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed planar vector fields the transition from small hopf type cycles to large relaxation cycles which occurs in an exponentially thin parameter interval is described as a perturbation of a family of singular cycles the results are obtained by means of two blow up transformations combined with standard tools of dynamical systems theory the efficient use of various charts is emphasized the results are applied to the van der pol equation

convergence of a weighted particle method for solving the boltzmann b g k equation we consider in this paper the numerical solution of the b g k model for the boltzmann equation the numerical method used here was introduced by mas gallic transport theory statist phys 16 1987 pp 885 887 using the results of perthame and pulvirenti arch rational mech anal 125 1993 pp 289 295 we first derive a uniqueness result for the b g k model and also regularity properties of the solution which are necessary for our analysis we then prove following mas gallic and poupaud transport theory statist phys 17 1988 pp 311 345 the convergence of the numerical approximation for the b g k model finally some numerical experiments are presented

classification of static plane symmetric space times according to their matter collineations in this paper we classify static plane symmetric space times according to their matter collineations these have been studied for both cases when the energy momentum tensor is nondegenerate and also when it is degenerate it turns out that the nondegenerate case yields either four five six seven or ten independent matter collineations in which four are isometries and the rest are proper there exists three interesting cases where the energy momentum tensor is degenerate but the group of matter collineations is finite dimensional the matter collineations in these cases are either four six or ten

anomalous diffusion limit induced on a kinetic equation using a compactness argument based on the velocity averaging lemma of golse et al it is shown that the limiting behavior of a kinetic linearized bgk gas model confined between two plates with maxwell boundary conditions when the distance between the plates goes to zero under a suitable anomalous scaling is diffusive we do not require the use of central limit theorems as in the method of borgers et al

smooth solutions to the bgk equation and the es bgk equation with infinite energy abstract the bgk model and the es bgk model of the boltzmann equation are of great importance in the kinetic theory of rarefied gases for the cauchy problems with general initial data smooth solutions were obtained only in bounded domains for both models in this paper we establish the existence and uniqueness of smooth solutions in the whole space more important some smooth solutions can permit infiniteness of energy

existence uniqueness and positivity of solutions for bgk models for mixtures abstract we consider kinetic models for a multi component gas mixture without chemical reactions in the literature one can find two types of bgk models in order to describe gas mixtures one type has a sum of bgk type interaction terms in the relaxation operator for example the model described by klingenberg pirner and puppo 20 which contains well known models of physicists and engineers for example hamel 16 and gross and krook 15 as special cases the other type contains only one collision term on the right hand side for example the well known model of andries aoki and perthame 1 for each of these two models 20 and 1 we prove existence uniqueness and positivity of solutions in the first part of the paper in the second part we use the first model 20 in order to determine an unknown function in the energy exchange of the macroscopic equations for gas mixtures described by dellacherie 11

global existence of classical solutions to the fokker planck bgk equation a kinetic model of the fokker planck boltzmann equation is introduced by replacing the original boltzmann collision operator with the bhatnagar gross krook collision model bgk collision model this model equation which we call the fokker planck bgk equation has many physical features that the fokker planck boltzmann equation possesses we first establish an l existence result for this equation by which we construct the approximate solutions then by means of the regularizing effects of the linear fokker planck operator and lp estimates of local maxwellians we obtain some uniform estimates of the approximate solutions finally combining those estimates and regularizing effects we prove by a compactness argument that the equation has a global classical solution under rather general initial conditions

global existence to the bgk model of boltzmann equation abstract we present an existence and a stability proof for solutions to the bgk model of boltzmann equation f v x f f m f t 0 x r n v r n m f p 2 t n 2 exp v u 2 2t p pu p u 2 pt 1 x r n 1 v v 2 f t x v dv it relies on the strong compactness of u t and on a new estimate on the third moment of v dv we also prove the entropy relation for 1

numerical passage from kinetic to fluid equations a numerical way to pass from the bhatnagar gross krook model of the boltzmann equation to compressible euler equations is presented in order to do so a stable discretization of the kinetic equati

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

cauchy problem for the ellipsoidal bgk model for polyatomic particles abstract we establish the existence and uniqueness of mild solutions for the polyatomic ellipsoidal bgk model which is a relaxation type kinetic model describing the evolution of polyatomic gaseous system at the mesoscopic level

collineations of the ricci tensor ricci collineations for the ricci tensor which is constructed from a general spherically symmetric and static metric are classified for all possibilities of rab r such that rab 0 for a b it turns out that the only collineations admitted by this tensor can be ten six or four and there does not appear any case in between

a bgk model for small prandtl number in the navier stokes approximation we present a bgk type collision model which approximates by a chapman enskog expansion the compressible navier stokes equations with a prandtl number that can be chosen arbitrarily between 0 and 1 this model has the basic properties of the boltzmann equation including theh theorem but contains an extra parameter in comparison with the standard bgk model this parameter is introduced multiplying the collision operator by a nonlinear functional of the distribution function it is adjusted to the prandtl number

a coupled ordinates method for solution acceleration of rarefied gas dynamics simulations non equilibrium rarefied flows are frequently encountered in a wide range of applications including atmospheric re entry vehicles vacuum technology and microscale devices rarefied flows at the microscale can be effectively modeled using the ellipsoidal statistical bhatnagar gross krook esbgk form of the boltzmann kinetic equation numerical solutions of these equations are often based on the finite volume method fvm in physical space and the discrete ordinates method in velocity space however existing solvers use a sequential solution procedure wherein the velocity distribution functions are implicitly coupled in physical space but are solved sequentially in velocity space this leads to explicit coupling of the distribution function values in velocity space and slows down convergence in systems with low knudsen numbers furthermore this also makes it difficult to solve multiscale problems or problems in which there is a large range of knudsen numbers in this paper we extend the coupled ordinates method comet previously developed to study participating radiative heat transfer to solve the esbgk equations in this method at each cell in the physical domain distribution function values for all velocity ordinates are solved simultaneously this coupled solution is used as a relaxation sweep in a geometric multigrid method in the spatial domain enhancements to comet to account for the non linearity of the esbgk equations as well as the coupled implementation of boundary conditions are presented the methodology works well with arbitrary convex polyhedral meshes and is shown to give significantly faster solutions than the conventional sequential solution procedure acceleration factors of 5 9 are obtained for low to moderate knudsen numbers on single processor platforms

mixed mode oscillations with multiple time scales mixed mode oscillations mmos are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes mmos have been observed and studied for over thirty years in chemical physical and biological systems few attempts have been made thus far to classify different patterns of mmos in contrast to the classification of the related phenomena of bursting oscillations this paper gives a survey of different types of mmos concentrating its analysis on mmos whose small amplitude oscillations are produced by a local multiple time scale mechanism recent work gives substantially improved insight into the mathematical properties of these mechanisms in this survey we unify diverse observations about mmos and establish a systematic framework for studying their properties numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper

numerical simulation of flows from free molecular regime to continuum regime by a dvm with streaming and collision processes a discrete velocity method dvm with streaming and collision processes is presented in this work for simulation of flows from free molecular regime to continuum regime the present scheme can be considered as a semi lagrangian like scheme at first we follow the conventional dvm to discretize the phase velocity space by a number of discrete velocities then for each discrete velocity the kinetic equation with bgk shakhov model is integrated in space and time within one time step as a result a simple algebraic formulation can be obtained and its solution can be marched in time by the streaming and collision processes however differently from the conventional semi lagrangian scheme the present scheme uses the muscl approach with van albada limiter in the process of reconstructing the distribution function at the surrounding points of the cell center and the transport distance is controlled in order to avoid extrapolation this makes the present scheme be capable of simulating the hypersonic rarefied flows in addition as compared to the unified gas kinetic scheme ugks the present scheme is simpler and easier for implementation thus the computational efficiency can be improved accordingly to validate the proposed numerical scheme test examples from free molecular regime to continuum regime are simulated numerical results showed that the present scheme can predict the flow properties accurately even for hypersonic rarefied flows

linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems abstract we are interested in high order linear multistep schemes for time discretization of adjoint equations arising within optimal control problems first we consider optimal control problems for ordinary differential equations and show loss of accuracy for adams moulton and adams bashforth methods whereas bdf methods preserve high order accuracy subsequently we extend these results to semi lagrangian discretizations of hyperbolic relaxation systems computational results illustrate theoretical findings

lp solutions to the cauchy problem of the bgk equation the bgk model of the boltzmann equation plays an important role in the kinetic theory of rarefied gases some existence and uniqueness results of solutions to its cauchy problem were established for large initial data under various circumstances see for example perthame b global existence to the bgk model of boltzmann equation j differ equations 82 191 205 1989 in this paper by establishing some weighted lp estimates of the hydrodynamical quantities of a gas we prove the existence theorem of the lp solutions to the cauchy problem and establish the propagation properties of the lp moments for this kind of solutions

regularity of the moments of the solution of a transport equation let u u x v satisfy the transport equation u v xu f x rn v rnwhere f belongs to some space of type lp dx d v where is a positive bounded measure on rn we study the resulting regularity of the moment u x v d v in terms of fractional sobolev spaces for example counter examples are given in order to test the optimality of our results