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

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

moment balancing templates for d k constrained codes and run length limited sequences the first order moment of d k constrained codes is investigated in this paper a generalized moment balancing template is proposed to encode a d k sequence into a single insertion or deletion correcting codeword without losing the constraint property by relocating 0 s in moment balancing runs which appear in a pairwise manner of a d k sequence the first order moment of this sequence can be modified to satisfy the varshamov tenengolts construction with a reasonably large base in the modulo system introduced by the varshamov tenengolts construction this generalized moment balancing template can be applied to run length limited sequences the asymptotic bound of the redundancy introduced by the template for d k sequences is of the same order as the universal template for random sequences and therefore the redundancy is small and suitable for long sequences of practical interest

traces of cm values of modular functions zagier proved that the traces of singular moduli i e the sums of the values of the classical j invariant over quadratic irrationalities are the fourier coefficients of a modular form of weight 3 2 with poles at the cusps using the theta correspondence we generalize this result to traces of cm values of weakly holomorphic modular functions on modular curves of arbitrary genus we also study the theta lift for the weight 0 eisenstein series for sl2 and realize a certain generating series of arithmetic intersection numbers as the derivative of zagier s eisenstein series of weight 3 2 this recovers a result of kudla rapoport and yang

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

theta lifts for lorentzian lattices and coefficients of mock theta functions we evaluate regularized theta lifts for lorentzian lattices in three different ways in particular we obtain formulas for their values at special points involving coefficients of mock theta functions by comparing the different evaluations we derive recurrences for the coefficients of mock theta functions such as hurwitz class numbers andrews spt function and ramanujan s mock theta functions

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

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

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

measure valued solutions to conservation laws motivated by traffic modelling motivated by a traffic model we study a conservation law whose solutions naturally need to be bounded radon measures we investigate the relations between this equation and finite systems of conservation laws that converge to it

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

an asymptotic preserving automatic domain decomposition method for the vlasov poisson bgk system with applications to plasmas in this work we present an efficient strategy to deal with plasma physics simulations in which localized departures from thermodynamical equilibrium are present the method is based on the introduction of intermediate regions which allows smooth transitions between kinetic and fluid zones in this paper we extend domain decomposition techniques obtained through dynamic coupling and buffer zones to the study of plasmas and moreover we combine them with asymptotic preserving and asymptotically accurate strategies for the time integration we use a hybrid scheme in which both kinetic and fluid descriptions are considered and coupled together while the kinetic model is solved by asymptotic preserving and accurate methods in order to guarantee high efficiency and accuracy in all regimes the numerical scheme is validated and its performances are analyzed by numerical simulations

harmonic maass forms of weight 1 the object of this paper is to initiate a study of the fourier coefficients of a weight one harmonic maass form and relate them to the complex galois representation associated to a weight one newform which is the form s image under a certain differential operator in this paper our focus will be on weight one dihedral newforms of prime level p 3 mod 4 in this case we give properties of the fourier coefficients that are similar to and sometimes reduce to cases of stark s conjectures on derivatives of l functions we also give a new modular interpretation of certain products of differences of singular moduli studied by gross and zagier finally we provide some numerical evidence that the fourier coefficients of a mock modular form whose shadow is exotic are similarly related to the associated complex galois representation

convergence of discrete velocity schemes for the boltzmann equation in this paper we prove the convergence of two discrete velocity deterministic schemes for the boltzmann equation namely buet s scheme and a new finite volume scheme that we introduce here we write the discretized equation in the form of a boltzmann continuous equation in order to be in the framework of the diperna lions theory of renormalized solutions in order to prove convergence we have to overcome two difficulties the convergence of the discretized collision kernel is very weak and the lemma on the compactness of velocity averages can be recovered only asymptotically when the parameter of discretization tends to zero

reliable communication over channels with insertions deletions and substitutions a new block code is introduced which is capable of correcting multiple insertion deletion and substitution errors the code consists of nonlinear inner codes which we call watermark codes concatenated with low density parity check codes over nonbinary fields the inner code allows probabilistic resynchronization and provides soft outputs for the outer decoder which then completes decoding we present codes of rate 0 7 and transmitted length 5000 bits that can correct 30 insertion deletion errors per block we also present codes of rate 3 14 and length 4600 bits that can correct 450 insertion deletion errors per block

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

a consistency result for a discrete velocity model of the boltzmann equation in this paper we study the link between a certain class of discrete velocity models dvms and the boltzmann equation those models possess an infinite number of velocities laying on a regular grid of step h in r3 our aim is to prove that it is possible to construct models consistent with the boltzmann equation i e such that the discrete collision term can be seen as an approximation of the collision integral of the boltzmann equation

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

the rationality of vector valued modular forms associated with the weil representation in a recent paper duke math j 97 219 233 borcherds asks whether or not the spaces of vector valued modular forms associated to the weil representation have bases of modular forms whose fourier expansions have only integer coefficients we give an affirmative answer to borcherds question this strengthens and simplifies borcherds main theorem which is a generalization of a theorem of gross kohnen and zagier math ann 278 497 562

on two geometric theta lifts the theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type in this paper we establish for the orthogonal group o p 2 an adjointness result between borcherds s singular theta lift and the kudla millson lift we extend this result to arbitrary signature by introducing a new singular theta lift for o p q on the geometric side this lift can be interpreted as a differential character in the sense of cheeger and simons for the cycles under consideration

heegner divisors l functions and harmonic weak maass forms recent works mostly related to ramanujan s mock theta functions make use of the fact that harmonic weak maass forms can be combinatorial generating functions generalizing works of waldspurger kohnen and zagier we prove that such forms also serve as generating functions for central values and derivatives of quadratic twists of weight 2 modular l functions to obtain these results we construct differentials of the third kind with twisted heegner divisor by suitably generalizing the borcherds lift to harmonic weak maass forms the connection with periods fourier coefficients derivatives of l functions and points in the jacobian of modular curves is obtained by analyzing the properties of these differentials using works of scholl waldschmidt and gross and zagier

a multiple insertion deletion correcting code for run length limited sequences a code construction is proposed to add a multiple insertion deletion error correcting capability to a run length limited sequence the codewords of this code are themselves run length limited the insertion deletion correcting capability is achieved by requiring several weighted sums of run lengths in the codewords to satisfy certain congruences modulo primes the construction is similar to the number theoretic code proposed by dolecek and anantharam which can correct multiple repetition errors or equivalently multiple insertions of zeros it is shown that if the codewords in this code are run length limited then the code is capable of correcting both insertions and deletions of zeros and ones an algorithm is proposed for decoding over a multiple insertion deletion channel following the work of dolecek and anantharam a systematic encoding method is also proposed for the codes furthermore it is shown that the proposed construction has a higher rate asymptotically than the helberg code which is unconstrained in terms of run lengths even though our construction has the additional run length constraints the need for run length limited codes that can correct insertion deletion errors is motivated by bit patterned media for magnetic recording

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

systematic encoding of the varshamov tenengol ts codes and the constantin rao codes the maximum number of information bits that can be encoded systematically by the number theoretic codes of varshamov and tenengol ts 1965 is determined this number is also studied for the more general class of the group theoretic constantin rao 1979 codes although these codes are at least as large as the varshamov tenengol ts codes it is shown that the number of bits that can be systematically encoded using a constantin rao code does not exceed the number of bits that can be systematically encoded using a varshamov tenengol ts code in fact in many cases the largest constantin rao code has the least number of bits that can be systematically encoded

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

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

moment balancing templates for spectral null codes the generalized moment balancing templates for spectral null codes are investigated in this paper a new approach based on the insertion of a determined number of vectors at determinate indices of a spectral null code word is found to balance the moment of the code word moment balanced code words preserve the spectral null properties and can guarantee the correction of one insertion or one deletion error as compared to the universal template for an uncoded sequence the redundancy is determined by the structure of the original spectral null code however it can be kept small the new approach involves the applications of the optimized moment balancing template and nonbinary enumeration of the balancing moment values

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

algebraic formulas for the coefficients of mock theta functions and weyl vectors of borcherds products abstract we present some applications of the kudla millson and the millson theta lift the two lifts map weakly holomorphic modular functions to vector valued harmonic maass forms of weight 3 2 and 1 2 respectively we give finite algebraic formulas for the coefficients of ramanujan s mock theta functions f q and q in terms of traces of cm values of a weakly holomorphic modular function further we construct vector valued harmonic maass forms whose shadows are unary theta functions and whose holomorphic parts have rational coefficients this yields a rationality result for the coefficients of mock theta functions i e harmonic maass forms whose shadows lie in the space of unary theta functions moreover the harmonic maass forms we construct can be used to evaluate the petersson inner products of unary theta functions with harmonic maass forms giving formulas and rationality results for the weyl vectors of borcherds products

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

rarefied flow computations using nonlinear model boltzmann equations high resolution finite difference schemes for solving the nonlinear model boltzmann equations are presented for the computations of rarefied gas flows the discrete ordinate method is first applied to remove the velocity space dependency of the distribution function which renders the model boltzmann equation in phase space to a set of hyperbolic conservation laws with source terms in physical space then a high order essentially nonoscillatory method due to harten et al j comput phys 71 231 1987 is adapted and extended to solve them explicit methods using operator splitting and implicit methods using the lower upper factorization are described to treat multidimensional problems the methods are tested for both steady and unsteady rarefied gas flows to illustrate its potential use the computed results using model boltzmann equations are found to compare well both with those using the direct simulation monte carlo results in the transitional regime flows and those with the continuum navier stokes calculations in near continuum regime flows

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

cm values of regularized theta lifts and harmonic weak maas forms of weight 1 we study special values of regularized theta lifts at complex multiplication cm points in particular we show that cm values of borcherds products can be expressed in terms of finitely many fourier coefficients of certain harmonic weak maas forms of weight 1 as it turns out these coefficients are logarithms of algebraic integers whose prime ideal factorization is determined by special cycles on an arithmetic curve our results imply a conjecture of duke and li and give a new proof of the modularity of a certain arithmetic generating series of weight 1 studied by kudla rapoport and yang

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

a variable length approach to moment balancing moment balancing templates have been proposed for channels with a small probability of an insertion deletion several orders smaller than additive errors that add a minimal amount of redundancy these templates are essentially a systematic way of encoding number theoretic codes primarily levenshtein s s 1 insertion deletion code moment balancing templates proposed up to this point have been of fixed length in this paper it is shown that by using variable length templates it is possible to obtain better performance than the optimal fixed length moment balancing template here performance is defined as the amount of redundancy which includes the moment balancing bits and the marker that needs to be added

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

moment balancing templates constructions to add insertion deletion correction capability to error correcting or constrained codes templates are constructed to extend arbitrary additive error correcting or constrained codes i e additional redundant bits are added in selected positions to balance the moment of the codeword the original codes may have error correcting capabilities or constrained output symbols as predetermined by the usual communication system considerations which are retained after extending the code using some number theoretic constructions in the literature insertion deletion correction can then be achieved if the template is carefully designed the number of additional redundant bits for the insertion deletion correction can be kept small in some cases of the same order as the number of parity bits in a hamming code of comparable length

faltings heights of cm cycles and derivatives of l functions we study the faltings height pairing of arithmetic special divisors and cm cycles on shimura varieties associated to orthogonal groups we compute the archimedean contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a rankin l function we prove the conjecture in certain cases where the shimura variety has dimension 0 1 or 2 in particular we obtain a new proof of the gross zagier formula

construction of fixed length insertion deletion correcting runlength limited codes an algorithm is presented for the construction of fixed length insertion deletion correcting runlength limited rll codes in this construction binary d k constrained codewords are generated by codewords of a q ary lee metric based code it is shown that this new construction always yields better codes than known constructions the use of a q ary lee 1987 metric code q odd is based on the assumption that an error insertion deletion or peak shift has maximal size q 1 2 it is shown that a decoding algorithm for the lee metric code can be extended so that it can also be applied to insertion deletion correcting rll codes furthermore such an extended algorithm can also correct some error patterns containing errors of size more than q 1 2 as a consequence if s denotes the maximal size of an error then in some cases the alphabet size of the generating code can be s 1 instead of 2 spl middot s 1

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

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

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

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

accuracy analysis of high order lattice boltzmann models for rarefied gas flows in this work we have theoretically analyzed and numerically evaluated the accuracy of high order lattice boltzmann lb models for capturing non equilibrium effects in rarefied gas flows in the incompressible limit the lb equation is shown to be able to reduce to the linearized bhatnagar gross krook bgk equation therefore when the same gauss hermite quadrature is used lb method closely resembles the discrete velocity method dvm in addition the order of hermite expansion for the equilibrium distribution function is found not to be directly correlated with the approximation order in terms of the knudsen number to the bgk equation for incompressible flows meanwhile we have numerically evaluated the lb models for a standing shear wave problem which is designed specifically for assessing model accuracy by excluding the influence of gas molecule surface interactions at wall boundaries the numerical simulation results confirm that the high order terms in the discrete equilibrium distribution function play a negligible role in capturing non equilibrium effect for low speed flows by contrast appropriate gauss hermite quadrature has the most significant effect on whether lb models can describe the essential flow physics of rarefied gas accurately our simulation results where the effect of wall gas interactions is excluded can lead to conclusion on the lb modeling capability that the models with higher order quadratures provide more accurate results for the same order gauss hermite quadrature the exact abscissae will also modestly influence numerical accuracy using the same gauss hermite quadrature the numerical results of both lb and dvm methods are in excellent agreement for flows across a broad range of the knudsen numbers which confirms that the lb simulation is similar to the dvm process therefore lb method can offer flexible models suitable for simulating continuum flows at the navier stokes level and rarefied gas flows at the linearized boltzmann model equation level