spectral lagrangian methods for collisional models of non equilibrium statistical states we propose a new spectral lagrangian based deterministic solver for the non linear boltzmann transport equation bte in d dimensions for variable hard sphere vhs collision kernels with conservative or non conservative binary interactions the method is based on symmetries of the fourier transform of the collision integral where the complexity in its computation is reduced to a separate integral over the unit sphere s d 1 the conservation of moments is enforced by lagrangian constraints the resulting scheme implemented in free space is very versatile and adjusts in a very simple manner to several cases that involve energy dissipation due to local micro reversibility inelastic interactions or elastic models of slowing down process our simulations are benchmarked with available exact self similar solutions exact moment equations and analytical estimates for the homogeneous boltzmann equation both for elastic and inelastic vhs interactions benchmarking of the simulations involves the selection of a time self similar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for maxwell molecules as studied first in bobylev et al a v bobylev c cercignani g toscani proof of an asymptotic property of self similar solutions of the boltzmann equation for granular materials journal of statistical physics 111 2003 403 417 and generalized to a wide range of related models in bobylev et al a v bobylev c cercignani i m gamba on the self similar asymptotics for generalized non linear kinetic maxwell models communication in mathematical physics in press url the method also produces accurate results in the case of inelastic diffusive boltzmann equations for hard spheres inelastic collisions under thermal bath where overpopulated non gaussian exponential tails have been conjectured in computations by stochastic methods t v noije m ernst velocity distributions in homogeneously cooling and heated granular fluids granular matter 1 57 1998 m h ernst r brito scaling solutions of inelastic boltzmann equations with over populated high energy tails journal of statistical physics 109 2002 407 432 s j moon m d shattuck j swift velocity distributions and correlations in homogeneously heated granular media physical review e 64 2001 031303 i m gamba s rjasanow w wagner direct simulation of the uniformly heated granular boltzmann equation mathematical and computer modelling 42 2005 683 700 and rigorously proven in gamba et al i m gamba v panferov c villani on the boltzmann equation for diffusively excited granular media communications in mathematical physics 246 2004 503 541 39 and a v bobylev i m gamba v panferov moment inequalities and high energy tails for boltzmann equations with inelastic interactions journal of statistical physics 116 2004 1651 1682

the milne and kramers problems for the boltzmann equation of a hard sphere gas existence uniqueness and properties of asymptotic behavior are proved for solutions of the milne and kramers problems for the linearized boltzmann equation for a gas of hard spheres the proof uses energy like estimates and follows the analysis of bardos santos and sentis for the equations of neutron transport

l p bernstein estimates and approximation by spherical basis functions the purpose of this paper is to establish l p error estimates a bernstein inequality and inverse theorems for approximation by a space comprising spherical basis functions located at scattered sites on the unit n sphere in particular the bernstein inequality estimates l p bessel potential sobolev norms of functions in this space in terms of the minimal separation and the l p norm of the function itself an important step in its proof involves measuring the l p stability of functions in the approximating space in terms of the l p norm of the coefficients involved as an application of the bernstein inequality we derive inverse theorems for sbf approximation in the l p norm finally we give a new characterization of besov spaces on the n sphere in terms of spaces of sbfs

application of the principle of minimizing the derivative to the construction of finite difference schemes for computing discontinuous solutions of gas dynamics progress in computer technology now allows the application of numerical methods to some practical multidimensional steady and unsteady gas flows while selecting a numerical method preference should be given to higher order schemes since they allow the study of the fine features of the solution and the reduction of the computation time in general however solutions of non stationary nonlinear equations of gas dynamics are not smooth and may include strong discontinuities such as shock waves or contact discontinuities this fact is important for choosing an appropriate numerical scheme if the strong discontinuities are extracted from the simulation domain and traced shock fitting while the rest of the flow is smooth it is possible to use numerical schemes with approximation close to the second order 1 3 however if the number of discontinuities increases with time leading to a complicated pattern the application of the shock fitting approach becomes difficult in this case it is preferable to apply shock capturing schemes 4 6 which have a lower order of approximation but allow the integration of the governing equations through the discontinuities without extracting their surfaces shock capturing schemes of a higher order of approximation are currently under development 7 9 in this paper certain aspects of constructing a numerical scheme applicable to discontinuous flows of gas dynamics are discussed using the idea of minimizing the derivatives of the solution a numerical scheme that is second order accurate in space and first order accurate in time is developed for the model equation u t u x 0 the properties of this scheme are analyzed and the conditions for its stability and monotonicity are formulated these results are then used in devising a similar scheme for the non stationary one dimensional equations of gas dynamics the performance of the new scheme is compared to that of other schemes using the break up of an initial discontinuity riemann problem as an example

fluid dynamic limits of kinetic equations i formal derivations the connection between kinetic theory and the macroscopic equations of fluid dynamics is described in particular our results concerning the incompressible navier stokes equations are based on a formal derivation in which limiting moments are carefully balanced rather than on a classical expansion such as those of hilbert or chapman enskog the moment formalism shows that the limit leading to the incompressible navier stokes equations like that leading to the compressible euler equations is a natural one in kinetic theory and is contrasted with the systematics leading to the compressible navier stokes equations some indications of the validity of these limits are given more specifically the connection between the diperna lions renormalized solution of the classical boltzmann equation and the leray solution of the navier stokes equations is discussed

derivation and analysis of near wall modelsfor channel and recirculating flows the problem of predicting features of turbulent flows occurs in many applications such as geophysical flows turbulent mixing pollution dispersal and even in the design of artificial hearts one promising approach is large eddy simulation les which seeks to predict local spatial averages u of the fluid s velocity u in some applications the les equations are solved over moderate time intervals and the core difficulty is associated with modeling near wall turbulence in complex geometries thus one important problem in les is to find appropriate boundary conditions for the flow averages which depend on the behavior of the unknown flow near the wall inspired by early works of navier and maxwell we develop such boundary conditions of the form email protected email protected email protected d email protected email protected email protected email protected email protected email protected email protected email protected email protected d email protected email protected 0 on the wall we derive effective friction coefficients b appropriate for both channel flows and recirculating flows and study their asymptotic behavior as the averaging radius s 0 and as the reynolds number re oc in the first limit no slip conditions are recovered in the second free slip conditions are recovered our goal herein is not to develop new theories of turbulent boundary layers but rather to use existing boundary layer theories to improve numerical boundary conditions for flow averages

velocity boundary condition at solid walls in rarefied gas calculations maxwell s famous slip boundary condition is often misapplied in current rarefied gas flow calculations e g in hypersonics microfluidics for simulations of gas flows over curved or moving surfaces this means crucial physics can be lost we give examples of such cases we also propose a higher order boundary condition based on maxwell s general equation and the constitutive relations derived by burnett unlike many other higher order slip conditions these are applicable to any form of surface geometry it is shown that these maxwell burnett boundary conditions are in reasonable agreement with the limited experimental data available for poiseuille flow and can also predict sone s thermal stress slip flow a phenomenon which cannot be captured by conventional slip boundary conditions

multidimensional upwind methods for hyperbolic conservation laws abstract we present a class of second order conservative finite difference algorithms for solving numerically time dependent problems for hyperbolic conservation laws in several space variables these methods are upwind and multidimensional in that the numerical fluxes are obtained by solving the characteristic form of the full multidimensional equations at the zone edge and that all fluxes are evaluated and differenced at the same time in particular operator splitting is not used correct behavior at discontinuities is obtained by the use of solutions to the riemann problem and by limiting some of the second order terms numerical results are presented which show that the methods described here yield the same high resolution as the corresponding operator split methods

tight pep lower bound for constellation rotated vector ofdm under carrier frequency offset and fast fading in this paper we study pairwise error probability pep of a system employing constellation rotated vector orthogonal frequency division multiplexing crv ofdm which is an enhancement of vector ofdm v ofdm with enhanced diversity over a frequency selective channel with both carrier frequency offset cfo and doppler frequency spread first we extend the work of rugini and banelli which is known to give a good bit error rate ber approximation for the standard ofdm system only with cfo to pep evaluation for crv ofdm and discuss its limitations then we discuss the conditionally gaussian characteristi2cs of the doppler induced interference and propose a conditionally gaussian pep approximation confirming a good match of the conditionally gaussian pep approximation and with simulation results we propose a new semianalytical pep lower bound the new lower bound is computationally feasible and is shown by simulation to be tight for small to moderate cfo finally a computationally more feasible approximation of the lower bound is also proposed our discussion reveals as a by product why the gaussian assumption for the doppler induced interference works for standard ofdm systems although the assumption is actually incorrect

axisymmetric lattice boltzmann method a lattice boltzmann method is developed for incompressible axisymmetric flows both force and source or sink terms are incorporated into the lattice boltzmann equation in a natural way which is consistent in dimension with the lattice boltzmann equation the correct macroscopic equations for incompressible axisymmetric flows are recovered through the chapman enskog expansion it turns out that the added terms are nothing but the additional in the governing equations for the axisymmetric flows compared with the navier stokes equations resulting in a simple and efficient model this provides an additional unique advantage that the proposed scheme is naturally suitable for general axisymmetric flows involving more physical phenomena two numerical simulations have been presented to verify the method

the kinetic scheme for the full burnett equations the present paper concerns two aspects for the burnett equations first we are going to theoretically show the consistency between the traditional chapman enskog expansion and the successive approximation for the bgk equation up to the super burnett order second we will design a numerical scheme to efficiently solve the burnett equations the current approach is an improvement of the bgk burnett scheme k xu phys fluids 15 2003 2077 where the locally constant collision frequency is considered based on the burnett distribution function a high order time accurate numerical flux is derived by different approaches the resulting scheme is tested in the problem of normal shock wave and that of force driven poiseuille flow

spherical marcinkiewicz zygmund inequalities and positive quadrature geodetic and meteorological data collected via satellites for example are genuinely scattered and not confined to any special set of points even so known quadrature formulas used in numerically computing integrals involving such data have had restrictions either on the sites points used or more significantly on the number of sites required here for the unit sphere embedded in r q we obtain quadrature formulas that are exact for spherical harmonics of a fixed order have nonnegative weights and are based on function values at scattered sites to be exact these formulas require only a number of sites comparable to the dimension of the space as a part of the proof we derive l 1 marcinkiewicz zygmund inequalities for such sites

matrix free interpolation on the sphere we study a subspace of bivariate trigonometric polynomials for interpolating functions on the sphere we give an explicit construction for a system of interpolation nodes and the corresponding basis for this space that allows a discrete fast fourier transform type formula for the interpolant we prove that the uniform norm of our interpolation operator is of the order log m2 where m is the number of interpolation points we also construct a minimal quadrature rule for our space with a number of points equal to the dimension of the space and describe an associated interpolation operator

blind estimation and detection of space time trellis coded transmissions over the rayleigh fading mimo channel we present a joint channel estimation and detection method of space time trellis codes sttc in the context of an unknown flat fading multiple input multiple output mimo channel a combined state space model for the space time code and the rayleigh fading mimo channel is introduced in order to use deterministic particle filtering at the receiver side an important feature of the proposed method is that the fading rate need not be known to the receiver monte carlo simulations show that the performances of the proposed scheme are close to decoding with perfect channel state information csi using the viterbi algorithm va

lattice boltzmann equation hydrodynamics by inserting position and time dependent source or forcing terms into the microscopic evolution equation of a lattice boltzmann fluid and treating the generalized scheme within the usual chapman enskog methodology we show that the emergent dynamics of the lattice fluid may be usefully transformed our method of adjustment is demonstrated by implementing the cylindrical polar coordinate form of the continuity and momentum equations on a rectangular lattice and generating results for pipe flow with straightforward systematic adjustment of the simulation our approach produces results in excellent agreement with theory

fast algorithms for discrete polynomial transforms consider the vandermonde like matrix p p k cos j n j k 0 n where the polynomials p k satisfy a three term recurrence relation if p k are the chebyshev polynomials t k then p coincides with c n 1 cos jk n j k 0 n this paper presents a new fast algorithm for the computation of the matrix vector product pa in o n log 2 n arithmetical operations the algorithm divides into a fast transform which replaces pa with c n 1 and a subsequent fast cosine transform the first and central part of the algorithm is realized by a straightforward cascade summation based on properties of associated polynomials and by fast polynomial multiplications numerical tests demonstrate that our fast polynomial transform realizes pa with almost the same precision as the clenshaw algorithm but is much faster for n 128

asymptotic analysis of a slightly rarefied gas with nonlocal boundary conditions in this paper nonlocal boundary conditions for the navier stokes equations are derived starting from the boltzmann equation in the limit for the knudsen number being vanishingly small in the same spirit of lombardo et al in j stat phys 130 69 82 2008 where a nonlocal poisson scattering kernel was introduced a gaussian scattering kernel which models nonlocal interactions between the gas molecules and the wall boundary is proposed it is proved to satisfy the global mass conservation and a generalized reciprocity relation the asymptotic expansion of the boundary value problem for the boltzmann equation provides in the continuum limit the navier stokes equations associated with a class of nonlocal boundary conditions of the type used in turbulence modeling

incompressible navier stokes and euler limits of the boltzmann equation we consider solutions of the boltzmann equation in a d dimensional torus d 2 3 for macroscopic times t n 1 t 0 when the space variations are on a macroscopic scale x n 1r n 2 x in the unit torus let u x t be for t t0 a smooth solution of the incompressible navier stokes equations ins for n 2 and of the incompressible euler equation ie for n 2 we prove that has solutions for t t0 which are close to o 2 in a suitable norm to the local maxwellian p 2 t d 2 exp v u x t 2 2t with constant density p and temperature t this is a particular case defined by the choice of initial values of the macroscopic variables of a class of such solutions in which the macroscopic variables satisfy more general hydrodynamical equations for n 3 these equations correspond to variable density ie while for n 2 they involve higher order derivatives of the density

analytical solutions of the lattice boltzmann bgk model analytical solutions of the two dimensional triangular and square lattice boltzmann bgk models have been obtained for the plane poiseuille flow and the plane couette flow the analytical solutions are written in terms of the characteristic velocity of the flow the single relaxation time and the lattice spacing the analytic solutions are the exact representation of these two flows without any approximation using the analytical solution it is shown that in poiseuille flow the bounce back boundary condition introduces an error of first order in the lattice spacing the boundary condition used by kadanoffet al in lattice gas automata to simulate poiseuille flow is also considered for the triangular lattice boltzmann bgk model an analytical solution is obtained and used to show that the boundary condition introduces an error of second order in the lattice spacing

inverted velocity profile in the cylindrical couette flow of a rarefied gas the cylindrical couette flow of a rarefied gas is investigated under the diffuse specular reflection condition of maxwell s type on the cylinders in the case where the inner cylinder is rotating whereas the outer cylinder is at rest the inverted velocity profile for small accommodation coefficients pointed out by tibbs baras and garcia phys rev e 56 2282 1997 on the basis of a monte carlo simulation is investigated extensively by means of a systematic asymptotic analysis for small knudsen numbers as well as the direct numerical analysis of the boltzmann equation and the parameter range in which the phenomenon appears is clarified

derivation of slip boundary conditions for the navier stokes system from the boltzmann equation rarefied gas flow behavior is usually described by the boltzmann equation the navier stokes system being valid when the gas is less rarefied slip boundary conditions for the navier stokes equations are derived in a rigorous and systematic way from the boundary condition at the kinetic level boltzmann equation these slip conditions are explicitly written in terms of asymptotic behavior of some linear half space problems the validity of this analysis is established in the simple case of the couette flow for which it is proved that the right boundary conditions are obtained

lower upper symmetric gauss seidel method for the euler and navier stokes equations on developpe un schema de relaxation multigrille application a des ecoulements transsoniques

polynomial operators and local approximation of solutions of pseudo differential equations on the sphere we study the solutions of an equation of the form lu f where l is a pseudo differential operator defined for functions on the unit sphere embedded in a euclidean space f is a given function and u is the desired solution we give conditions under which the solution exists and deduce local smoothness properties of u given corresponding local smoothness properties of f measured by local besov spaces we study the global and local approximation properties of the spectral solutions describe a method to obtain approximate solutions using values of f at points on the sphere and polynomial operators and describe the global and local rates of approximation provided by our polynomial operators

analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice boltzmann bgk model in this paper we analytically solve the velocity of the lattice boltzmann bgk equation lbgk for several simple flows the analysis provides a framework to theoretically analyze various boundary conditions in particular the analysis is used to derive the slip velocities generated by various schemes for the nonslip boundary condition we find that the slip velocity is zero as long as f e 0 at boundaries no matter what combination of distributions is chosen the schemes proposed by nobleet al and by inamuroet al yield the correct zeroslip velocity while some other schemes such as the bounce back scheme and the equilibrium distribution scheme would inevitably generate a nonzero slip velocity the bounce back scheme with the wall located halfway between a flow node and a bounce back node is also studied for the simple flows considered and is shown to produce results of second order accuracy the momentum exchange at boundaries seems to be highly related to the slip velocity at boundaries to be specific the slip velocity is zero only when the momentum dissipated by boundaries is equal to the stress provided by fluids

rotationally invariant quadratures for the sphere we construct near optimal quadratures for the sphere that are invariant under the icosahedral rotation group these quadratures integrate all n 1 2 linearly independent functions in a rotationally invariant subspace of maximal order and degree n the nodes of these quadratures are nearly uniformly distributed and the number of nodes is only marginally more than the optimal n 1 2 3 nodes using these quadratures we discretize the reproducing kernel on a rotationally invariant subspace to construct an analogue of lagrange interpolation on the sphere this representation uses function values at the quadrature nodes in addition the representation yields an expansion that uses a single function centred and mostly concentrated at nodes of the quadrature thus providing a much better localization than spherical harmonic expansions we show that this representation may be localized even further we also describe two algorithms of complexity for using these grids and representations finally we note that our approach is also applicable to other discrete rotation groups

instabilities in the chapman enskog expansion and hyperbolic burnett equations it is well known that the classical chapman enskog procedure does not work at the level of burnett equations the next step after the navier stokes equations roughly speaking the reason is that the solutions of higher equations of hydrodynamics burnett s etc become unstable with respect to short wave perturbations this problem was recently attacked by several authors who proposed different ways to deal with it we present in this paper one of possible alternatives first we deduce a criterion for hyperbolicity of burnett equations for the general molecular model and show that this criterion is not fulfilled in most typical cases then we discuss in more detail the problem of truncation of the chapman enskog expansion and show that the way of truncation is not unique the general idea of changes of coordinates based on analogy with the theory of dynamical systems leads finally to nonlinear hyperbolic burnett equations hbes without using any information beyond the classical burnett equations it is proved that hbes satisfy the linearized h theorem the linear version of the problem is studied in more detail the complete chapman enskog expansion is given for the linear case a simplified proof of the slemrod identity for burnett coefficients is also given

implicit high order method for calculating rarefied gas flow in a planar microchannel an efficient numerical algorithm for calculating rarefied gas flows in planar microchannels on the basis of the boltzmann kinetic equation with the linearized s model collision integral is presented the algorithm consists of a high order spatial discretisation on unstructured meshes conservative procedure to calculate macroscopic quantities and efficient one step implicit time evolution method it therefore works across all flow regimes from the free molecular to nearly continuum ones and provides rapid convergence to steady state solutions the parallel implementation is provided via mpi programming paradigm the performance of the method is illustrated by computing the flow for length to the width ratios for up to 10 3 in the wide range of knudsen numbers numerical estimates of efficiency of implicit time marching and parallel performance as well as convergence studies in both physical and velocity spaces are provided the end effects and other essentially two dimensional features of the flow are analysed and a detailed comparison with the case of an infinitely long channel is carried out the presented numerical data can be used as reference for future studies of nonlinear flows as well as for comparison with other numerical approaches and flow models

consistent lattice boltzmann modeling of low speed isothermal flows at finite knudsen numbers in slip flow regime ii application to curved boundaries gaseous flows inside microfluidic devices often fall in the slip flow regime according to this theoretical description the navier stokes model remains applicable in bulk while at solid walls a slip velocity boundary model shall be considered physically it is well established that to properly account for the wall curvature the wall slip velocity must be determined by the shear stress rather than the normal component of the velocity derivative alone as commonly applied to planar surfaces it follows that the numerical transcription of this type of boundary condition is generally a challenging task for standard computational fluid dynamics cfd techniques this paper aims to show that the realization of the slip velocity condition on arbitrarily shaped boundaries can be accomplished in a natural way with the lattice boltzmann method lbm to substantiate this conclusion this work undertakes the following three studies first we examine the conditions under which the generic reflection type boundary rules used by lbm become consistent models for the slip velocity boundary condition this effort makes use of the second order chapman enskog expansion method where we address both planar and curved boundaries the analysis also clarifies the capabilities and limitations behind the considered reflection type slip schemes second we revisit the family of parabolic accurate lbm slip boundary schemes originally formulated in phys rev e 96 013311 2017 2470 004510 1103 physreve 96 013311 on the basis of the multireflection framework and discuss their characteristics when operating on curved boundaries as well as the limitations of other less accurate lbm slip boundary formulations such as the linearly accurate slip schemes and the widely popular kinetic based boundary schemes in addition we also discuss the numerical stability of the parabolic slip schemes previously developed providing an heuristic strategy to improve their stable range of operation third we evaluate the performance of the several slip boundary schemes debated in this paper the numerical tests correspond to two classical 2d benchmark flow problems of slip over non planar solid surfaces namely i the velocity profile of the cylindrical couette flow and ii the permeability of a slow rarefied gas over a periodic array of circular cylindrical obstacles the obtained numerical results confirm the competitiveness of the lbm when equipped with slip boundary schemes of parabolic accuracy as cfd tool to simulate slippage phenomena over arbitrarily non planar surfaces indeed although operating on a simple uniform mesh discretization the lbm yields a similar or even superior level of accuracy compared to state of the art fem simulations conducted on hardworking body fitted meshes this conclusion establishes the lbm as a very appealing cfd technique for simulating microfluidic flows in the slip flow regime a result that deserves further exploration in future studies

the simulation of independent rayleigh faders multiple independent rayleigh fading waveforms are often required for the simulation of wireless communications channels jakes 1974 rayleigh fading model and its derivatives based on the sum of sinusoids provide simple simulators but they have major shortcomings in their simulated correlation functions a novel sum of sinusoids fading model is proposed and verified which generates rayleigh fading processes satisfying the theoretical independence requirements and providing desired power spectral densities with ideal second order moment the effects of replacing sinusoids in the proposed model by their approximate waveforms are also analyzed and tested performance evaluation and comparison are provided using the quality measures of the mean square error of autocorrelation function and the second order moment of the power spectral density

minimum sobolev norm interpolation of scattered derivative data abstract we study the problem of reconstructing a function on a manifold satisfying some mild conditions given data of the values and some derivatives of the function at arbitrary points on the manifold while the problem of finding a polynomial of two variables with total degree n given the values of the polynomial and some of its derivatives at exactly the same number of points as the dimension of the polynomial space is sometimes impossible we show that such a problem always has a solution in a very general situation if the degree of the polynomials is sufficiently large we give estimates on how large the degree should be and give explicit constructions for such a polynomial even in a far more general case as the number of sampling points at which the data is available increases our polynomials converge to the target function on the set where the sampling points are dense numerical examples in single and double precision show that this method is stable efficient and of high order

modified lattice boltzmann model for axisymmetric flows a modified lattice boltzmann model based on the two dimensional nine velocity lattice bhatnagar gross krook fluid is presented for axisymmetric flows a spatially and temporally varying source term is incorporated into the evolution equation for the momentum distribution function on a two dimensional cartesian lattice the precise form of the source term is derived through a chapman enskog analysis so that the additional axisymmetric contributions in the navier stokes equations are furnished when written in the cylindrical polar coordinate system

localized linear polynomial operators and quadrature formulas on the sphere the purpose of this paper is to construct universal auto adaptive localized linear polynomial valued operators based on scattered data on the hyper sphere mathbb s q q ge2 the approximation and localization properties of our operators are studied theoretically in deterministic as well as probabilistic settings numerical experiments are presented to demonstrate their superiority over traditional least squares and discrete fourier projection polynomial approximations an essential ingredient in our construction is the construction of quadrature formulas based on scattered data exact for integrating spherical polynomials of moderately high degree our formulas are based on scattered sites i e in contrast to such well known formulas as driscoll healy formulas we need not choose the location of the sites in any particular manner while the previous attempts to construct such formulas have yielded formulas exact for spherical polynomials of degree at most 18 we are able to construct formulas exact for spherical polynomials of degree 178

simulation models with correct statistical properties for rayleigh fading channels in this paper new sum of sinusoids statistical simulation models are proposed for rayleigh fading channels these new models employ random path gain random initial phase and conditional random doppler frequency for all individual sinusoids it is shown that the autocorrelations and cross correlations of the quadrature components and the autocorrelation of the complex envelope of the new simulators match the desired ones exactly even if the number of sinusoids is as small as a single digit integer moreover the probability density functions of the envelope and phase the level crossing rate the average fade duration and the autocorrelation of the squared fading envelope which contains fourth order statistics of the new simulators asymptotically approach the correct ones as the number of sinusoids approaches infinity while good convergence is achieved even when the number of sinusoids is as small as eight the new simulators can be directly used to generate multiple uncorrelated fading waveforms for frequency selective fading channels multiple input multiple output channels and diversity combining scenarios statistical properties of one of the new simulators are evaluated by numerical results finding good agreements

the generation of correlated rayleigh random variates by inverse discrete fourier transform a number of different algorithms are used for the generation of correlated rayleigh random variates this paper presents an analysis of the statistical properties of methods based on the inverse discrete fourier transform idft a modification of the algorithm of smith 1975 is presented the new method requiring exactly one half the number of idpt operations and roughly two thirds the computer memory of the original method evaluations of and comparisons between various variate generation methods using meaningful quantitative measures are believed to be lacking new quantitative quality measures for random variate generation have been proposed that are in particular meaningful and useful for digital communication system simulation this paper presents the application of these measures to the idft method and three other methods of correlated variate generation comparing the algorithms in terms of the quality of the generated samples and the required computational effort

on the multidimensional gas kinetic bgk scheme this paper concerns the performance difference among the multidimensional md gas kinetic bgk scheme the corresponding quasi one dimensional q1d extension and the directional splitting ds scheme with kinetic boundary conditions the md scheme includes tangential slopes in the flux calculation which are absent in a q1d or ds method in spite of taking more computational time the md scheme is found to be able to capture the characteristics of rarefied flow having a boundary with curvature or a nonuniform temperature such as the inverted velocity distribution in rarefied cylindrical couette flow and the weak flow field induced by the temperature gradient of a body where the q1d fails the md scheme can also yield clearly better results in high speed microchannel flow and power law fluid flow between concentric rotating cylinders when compared with direct simulation monte carlo studies and analytic solutions in the low reynolds number flow around a naca0012 airfoil case the md scheme and the q1d method predict wall heat flux distribution with obvious difference the ds method predicts results nearly identical to those from q1d in these steady flow cases except that it can hardly give reasonable solutions in power law fluid case with viscous exponential factor away from unity correct prediction of the stress and the wall temperature gradient is considered responsible for this better performance in simulations of the scalar convection diffusion equation it is found that the inclusion of tangential slopes in the flux computation can clearly improve the temporal accuracy of the scheme in this case the ds scheme also shows good performance the present study suggests caution in adopting of the q1d extension or ds method in multidimensional flow when it is sensitive to the accuracy of stress or wall variable gradient calculations it is better to use the md scheme for high performance simulation

analytic solutions of linearized lattice boltzmann equation for simple flows a general procedure to obtain analytic solutions of the linearized lattice boltzmann equation for simple flows is developed as examples the solutions for the poiseuille and the plane couette flows in two dimensional space are obtained and studied in detail the solutions not only have a component which is the solution of the navier stokes equation they also include a kinetic component which cannot be obtained by the navier stokes equation the kinetic component of the solutions is due to the finite mean free path effect comparison between the analytic results and the numerical results of lattice gas simulations is made and they are found to be in accurate agreement

interpolation of bounded bandlimited signals and applications a method to interpolate a bounded bandlimited signal from its own samples with minimum complexity is presented that guarantees an a priori specified accuracy the method is the result of combining a fast converging sampling expansion with the farrow interpolator technique it provides important complexity reductions in various signal processing applications from which three cases are studied for the farrow interpolator it assures that the interpolation error will be smaller than a known bound both in the time and frequency domains for delay estimation the method allows one to decouple a given estimation detection algorithm in two steps in the first one a few finite convolutions are carried out in order to compute a set of interpolation coefficients in the second the algorithm is executed but with a complexity independent of the length of the temporal observation interval besides it is shown how to introduce this decoupling in the matched filter music and conditional maximum likelihood delay estimators finally the method is employed to give a simple solution to an important efficiency problem in the simulation of communications systems the generation of rayleigh processes

boundary singularity for thermal transpiration problem of the linearized boltzmann equation we study the boundary singularity of the fluid velocity for the thermal transpiration problem in the kinetic theory logarithmic singularity has been demonstrated through the asymptotic and computational analysis the goal of this paper is to confirm this logarithmic singularity through exact analysis we use an iterated scheme with the gain part of the collision operator as a source the iterated scheme is appropriate for large knudsen numbers considered here and yields an explicit leading term

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

non local scattering kernel and the hydrodynamic limit in this paper we study the interaction of a fluid with a wall in the framework of the kinetic theory we consider the possibility that the fluid molecules can penetrate the wall to be reflected by the inner layers of the wall this results in a scattering kernel which is a non local generalization of the classical maxwell scattering kernel the proposed scattering kernel satisfies a global mass conservation law and a generalized reciprocity relation we study the hydrodynamic limit performing a knudsen layer analysis and derive a new class of weakly nonlocal boundary conditions to be imposed to the navier stokes equations

high order numerical methods for the space non homogeneous boltzmann equation in this paper we present accurate methods for the numerical solution of the boltzmann equation of rarefied gas the methods are based on a time splitting technique the transport is solved by a third order accurate in space positive and flux conservative pfc method the collision step is treated by a fourier approximation of the collision integral which guarantees spectral accuracy in velocity coupled with several high order integrators in time strang splitting is used to achieve second order accuracy in space and time several numerical tests illustrate the properties of the methods

anisotropic scattering kernel generalized and modified maxwell boundary conditions this article presents a model of a scattering kernel of boundary conditions for the boltzmann equation the proposed scattering kernel is based on an anisotropic accommodation argument three parameters equal to the momentum accommodation coefficients are shown as characterizing the influence of each direction first the new scattering kernel is derived from a phenomenological criticism of the first form of the scattering kernel proposed by maxwell then the same result is established from an analytic approach based on the spectral nature of the linear integral operator associated to the scattering kernel problem as a result the model provides a correct form of scattering kernel to handle the influence of each direction in particle collisions with the wall finally independent accommodation of each internal mode is added to extend the model to the case of polyatomic gases

second order knudsen layer analysis for the generalized slip flow theory ii curvature effects numerical analyses of the second order knudsen layer are carried out on the basis of the linearized boltzmann equation for hard sphere molecules under the diffuse reflection boundary condition the effects of the boundary curvature have been clarified in details thereby completing the numerical data required up to the second order of the knudsen number for the asymptotic theory of the boltzmann equation the generalized slip flow theory a local singularity appears as a result of the expansion at the level of the velocity distribution function when the curvature exists

kinetic boundary conditions in the lattice boltzmann method derivation of the lattice boltzmann method from the continuous kinetic theory x he and l s luo phys rev e 55 r6333 1997 x shan and x he phys rev lett 80 65 1998 is extended in order to obtain boundary conditions for the method for the model of a diffusively reflecting moving solid wall the boundary condition for the discrete set of velocities is derived and the error of the discretization is estimated numerical results are presented which demonstrate convergence to the hydrodynamic limit in particular the knudsen layer in the kramers problem is reproduced correctly for small knudsen numbers

slip boundary conditions for the compressible navier stokes equations the slip boundary conditions for the compressible navier stokes equations are derived systematically from the boltzmann equation on the basis of the chapman enskog solution of the boltzmann equation and the analysis of the knudsen layer adjacent to the boundary the resulting formulas of the slip boundary conditions are summarized with explicit values of the slip coefficients for hard sphere molecules as well as the bhatnagar gross krook model these formulas which can be applied to specific problems immediately help to prevent the use of often used slip boundary conditions that are either incorrect or without theoretical basis

accuracy of the lattice boltzmann method based on analytical solutions in this paper a simple method is proposed to obtain steady analytical solutions for the lattice boltzmann method based on such analytical results it is demonstrated how the accuracy of the lattice boltzmann method can depend on the relative orientation of the lattice and the flow field it is also demonstrated that the method can be useful to obtain a general class of analytical solutions for the lattice boltzmann method finally a simple relation is given between the compressibility error and the velocity field

analytical solution of axi symmetrical lattice boltzmann model for cylindrical couette flows abstract analytical solution for the axi symmetrical lattice boltzmann model is obtained for the low mach number cylindrical couette flows in the hydrodynamic limit the present solution is in excellent agreement with the result of the navier stokes equation since the kinetic boundary condition is used the present analytical solution using nine discrete velocities can describe flows with the knudsen number up to 0 1 meanwhile the comparison with the simulation data obtained by the direct simulation monte carlo method shows that higher order lattice boltzmann models with more discrete velocities are needed for highly rarefied flows

unified solver for rarefied and continuum flows with adaptive mesh and algorithm refinement this paper describes a unified flow solver ufs for rarefied and continuum gas flows the ufs separates the rarefied and continuum flow domains and selects appropriate solvers to combine the efficiency of continuum models with the accuracy of kinetic models the direct numerical solution of the boltzmann transport equation is used in rarefied regions while kinetic schemes of continuum fluid dynamics are used elsewhere using similar computational techniques for the kinetic and continuum solvers and employing intelligent domain decomposition algorithms attain the efficiency and numerical stability of the ufs solutions of test problems are presented to illustrate the capabilities of the ufs for high and low speed flows it is shown that the ufs can dynamically adapt the computational mesh and automatically introduce and remove kinetic patches to provide significant savings by limiting molecular scale solutions only to the regions where they are needed

application of gas kinetic scheme with kinetic boundary conditions in hypersonic flow this work focuses on the application of the gas kinetic scheme based on the bhatnagar gross krook particle collision model in the hypersonic flow simulations kinetic boundary conditions with different accommodation coefficients are constructed and implemented in the scheme the numerical study is based on a laminar shock wave boundary layer interaction in a mach 10 flow passing through a hollow cylinder flare model the grid independent numerical results agree well with experimental measurements the utilization of the kinetic slip boundary condition is necessary to improve the agreement between the current results experimental measurements as well as direct simulation monte carlo solutions the effect of the variation of the accommodation coefficient on the flow solutions is quantitatively evaluated

factorization symmetry in the lattice boltzmann method abstract a non perturbative algebraic theory of the lattice boltzmann method is developed based on the symmetry of a product it involves three steps i derivation of admissible lattices in one spatial dimension through a matching condition which imposes restricted extension of higher order gaussian moments ii a special quasi equilibrium distribution function found analytically in closed form on the product lattice in two and three spatial dimensions and which proves the factorization of quasi equilibrium moments and iii an algebraic method of pruning based on a one into one relation between groups of discrete velocities and moments two routes of constructing lattice boltzmann equilibria are distinguished the present theory includes previously known limiting and special cases of lattices and enables automated derivation of lattice boltzmann models from two dimensional tables by finding the roots of one polynomial and solving a few linear systems

asymptotic theory for the time dependent behavior of a slightly rarefied gas over a smooth solid boundary a time evolution of a slightly rarefied monoatomic gas namely a gas for small knudsen numbers which is perturbed slowly and slightly from a reference uniform equilibrium state at rest is investigated on the basis of the linearized boltzmann equation by a systematic asymptotic analysis a set of fluid dynamic type equations and its boundary conditions that describe the gas behavior up to the second order of the knudsen number are derived the developed theory covers a general intermolecular potential and a gas surface interaction it is shown that i the compressibility of the gas manifests itself from the leading order in the energy equation and from the first order in the continuity equation ii although the momentum equation is the stokes equation it contains a double laplacian of the leading order flow velocity as a source term at the second order iii a double laplacian source term also appears in the energy equation at the second order iv the slip and jump conditions are the same as those in the time independent case up to the first order and the difference occurs at the second order in the jump conditions as the terms of the divergence of flow velocity and of the laplacian of temperature numerical values of all the slip and jump coefficients are obtained for a hard sphere gas by the use of a symmetric relation developed recently

corrigendum to spherical marcinkiewicz zygmund inequalities and positive quadrature professor dr joachim stockler has kindly pointed out to us that our proofs 2 of the estimates 4 5 and 4 9 are valid only in the case when we do not require the quadrature weights a to be nonnegative since the krein rutman extensions of nonnegative functionals are not guaranteed to be norm preserving the purpose of this note is to point out that the existence of nonnegative weights a satisfying 4 4 necessarily implies the following analogue 0 1 of 4 5 the existence of such weights was proved in the paper here and in the sequel we use all the notation as in the paper a q r c p c 1 p 0 1

on the construction of kinetic schemes a general method for the construction of kinetic schemes of evolutionary equations is illustrated with the simple example of the linear advection equation where the role of the collision effect is clarified theoretically and numerically the application to the compressible euler equation and navier stokes equation is explained the theoretical backgrounds of the existing kinetic schemes such as the pullin scheme for the euler equation and chou baganoff scheme and modified prendergast xu scheme for the navier stokes equation are unified and their intrinsic errors are revealed

from the boltzmann equation to the euler equations in the presence of boundaries the fluid dynamic limit of the boltzmann equation leading to the euler equations for an incompressible fluid with constant density in the presence of material boundaries shares some important features with the better known inviscid limit of the navier stokes equations the present paper slightly extends recent results from c bardos f golse l paillard the incompressible euler limit of the boltzmann equation with accommodation boundary condition comm math sci 10 2012 159 190 to the case of boundary conditions for the boltzmann equation more general than maxwell s accommodation condition

anomalous flow profile due to the curvature effect on slip length einzel panzer and liu phys rev lett 64 2269 1990 suggest that the slip boundary condition for a fluid moving near a wall is modified by the radius of curvature of the surface using particle simulations of a microscopic flow between concentric cylinders we qualitatively confirm their prediction and point out that the effect is seen in a limiting case derived by maxwell nature london 16 244 1877

discrete velocity models and numerical schemes for the boltzmann bgk equation in plane and axisymmetric geometries we present new numerical models for computing transitional or rarefied gas flows as described by the boltzmann bgk and bgk es equations we first propose a new discrete velocity model based on the entropy minimization principle this model satisfies the conservation laws and the entropy dissipation moreover the problem of conservation and entropy for axisymmetric flows is investigated we find algebraic relations that must be satisfied by the discretization of the velocity derivative appearing in the transport operator then we propose some models that satisfy these constraints owing to these properties we obtain numerical schemes that are economic in terms of discretization and robust in particular we develop a linearized implicit scheme for computing stationary solutions of the discrete velocity bgk and bgk es models this scheme is the basis of a code which can compute high altitude hypersonic flows in 2d plane and axisymmetric geometries our results are analyzed and compared to other methods

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