semiautomatic construction of lattice boltzmann models a crucial step in constructing a lattice boltzmann model is the definition of a suitable set of lattice velocities and the correct assignment of the associated weights for high order models the solution of this problem requires a nontrivial effort this paper outlines the functioning of a publicly available python script which has been written to assist researchers in that task the speed of sound c s is considered as a parameter which can within limits be chosen at will under this premise the maxwell boltzmann constraint equations are a system of linear equations to determine the weights and hence amenable to numerical solution by standard linear algebra library routines by suitable contractions the tensor equations are mapped to a set of equivalent scalar equations which simplifies the treatment significantly for a user supplied set of velocity shells the software first checks if a solution for the weights exists and returns it if it also happens to be unique in such a case the software also calculates the range of c s values that yield positive weights standard models like d3q19 with a well defined special c s value then result as limiting cases where one of the weights vanishes in the case of an infinite set of solutions the user may find one particular solution by supplying a c s value and then minimizing one or several weights within the framework of standard linear programming some examples illustrate the feasibility and usefulness of the approach a number of models that have been discussed in the literature are nicely reproduced while the software has also been able to find some new models of even higher order

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

multiphase cascaded lattice boltzmann method to improve the stability of the lattice boltzmann method lbm at high reynolds number the cascaded lbm has recently been introduced as in the multiple relaxation time mrt method the cascaded lbm introduces additional relaxation times into the collision operator but does so in a co moving reference frame this has been shown to significantly increase stability at low viscosity in the single phase case here the cascaded lbm is further developed to include multiphase flow for this the force term is calculated by the interaction potential method and introduced into the collision operator via the exact difference method edm comparisons are made with the lattice bhatnagar gross krook lbgk method and an mrt implementation both the cascaded and mrt methods are shown to significantly reduce spurious velocities over the lbgk method for the particular case of the shan chen interparticle force term calculation with the edm the cascaded lbm is successfully combined with a multiphase method and shown to perform as well as the more established mrt method the cascaded lbm is found to be a considerably improved approach to the simulation of multiphase flow over the lbgk significantly increasing the stability range of both density ratio and reynolds number additionally the importance of including third order velocity terms in the equilibria of both the cascaded and mrt methods is discussed

connection between kinetic methods for fluid dynamic equations and macroscopic finite difference schemes the lattice boltzmann method lbm for the incompressible navier stokes ns equations and the gas kinetic scheme for the compressible ns equations are based on the kinetic theory of gases in the latter case however it is shown that the kinetic formulation is necessary only in the discontinuous reconstruction of fluid dynamic variables for shock capturing analogously we will discuss the reduction of a kinetic method for the incompressible case where the lbm scheme will be shown to shrink to an artificial compressibility type finite difference scheme we will prove first that a simple and compact lbm scheme cannot catch rarefied effects beyond navier stokes and hence that it is worth the effort to develop kinetic based fd alternatives finally we will propose two improvements to existing kinetic based fd schemes first of all a the proposed scheme is formulated purely in terms of macroscopic quantities on a compact stencil secondly b the semi implicit formulation is proposed in order to increase the stability we think that this work may be useful to others in realizing the actual possibilities of simple lbm schemes beyond navier stokes and in adopting the suggested improvements in their actual fd codes

truncation errors and the rotational invariance of three dimensional lattice models in the lattice boltzmann method abstract the application of the lattice boltzmann method lbm in three dimensional isothermal hydrodynamic problems often adopts one of the following models d3q15 d3q19 or d3q27 although all of them retrieve consistent navier stokes dynamics in the continuum limit they are expected to behave differently at discrete level the present work addresses this issue by performing a lbm truncation error analysis as a conclusion it is theoretically demonstrated that differences among the aforementioned cubic lattices lie in the structure of their non linear truncation errors while reduced lattice schemes such as d3q15 and d3q19 introduce spurious angular dependencies through non linear truncation errors the complete three dimensional cubic lattice d3q27 is absent from such features this result justifies the superiority of the d3q27 lattice scheme to cope with the rotational invariance principle in three dimensional isothermal hydrodynamic problems particularly when convection is not negligible such a theoretical conclusion also finds support in numerical tests presented in this work a poiseuille duct flow and a weakly rotating duct flow

lattice wigner equation we present a numerical scheme to solve the wigner equation based on a lattice discretization of momentum space the moments of the wigner function are recovered exactly up to the desired order given by the number of discrete momenta retained in the discretization which also determines the accuracy of the method the wigner equation is equipped with an additional collision operator designed in such a way as to ensure numerical stability without affecting the evolution of the relevant moments of the wigner function the lattice wigner scheme is validated for the case of quantum harmonic and anharmonic potentials showing good agreement with theoretical results it is further applied to the study of the transport properties of one and two dimensional open quantum systems with potential barriers finally the computational viability of the scheme for the case of three dimensional open systems is also illustrated

kinetic solvers with adaptive mesh in phase space an adaptive mesh in phase space amps methodology has been developed for solving multidimensional kinetic equations by the discrete velocity method a cartesian mesh for both configuration r and velocity v spaces is produced using a tree of trees tot data structure the r mesh is automatically generated around embedded boundaries and is dynamically adapted to local solution properties the v mesh is created on the fly in each r cell mappings between neighboring v space trees is implemented for the advection operator in r space we have developed algorithms for solving the full boltzmann and linear boltzmann equations with amps several recent innovations were used to calculate the discrete boltzmann collision integral with dynamically adaptive v mesh the importance sampling multipoint projection and variance reduction methods we have developed an efficient algorithm for calculating the linear boltzmann collision integral for elastic and inelastic collisions of hot light particles in a lorentz gas our amps technique has been demonstrated for simulations of hypersonic rarefied gas flows ion and electron kinetics in weakly ionized plasma radiation and light particle transport through thin films and electron streaming in semiconductors we have shown that amps allows minimizing the number of cells in phase space to reduce the computational cost and memory usage for solving challenging kinetic problems

univariate polynomial equation providing on lattice higher order models of thermal lattice boltzmann theory a univariate polynomial equation is presented it provides on lattice higher order models of the thermal lattice boltzmann equation the models can be accurate up to any required level and can be applied to regular lattices which allow efficient and accurate approximate solutions of the boltzmann equation we derive models approaching the complete galilean invariant and providing accuracy of the fourth order moment and beyond we simulate one dimensional thermal shock tube problems to illustrate the accuracy of our models moreover we show the remarkably enhanced stability obtained by our models and our discretized equilibrium distributions

lattices for the lattice boltzmann method a recently introduced theory of higher order lattice boltzmann models chikatamarla and karlin phys rev lett 97 190601 2006 is elaborated in detail a general theory of the construction of lattice boltzmann models as an approximation to the boltzmann equation is presented new lattices are found in all three dimensions and are classified according to their accuracy degree of approximation of the boltzmann equation the numerical stability of these lattices is argued based on the entropy principle the efficiency and accuracy of many new lattices are demonstrated via simulations in all three dimensions

general solution of lattices for cartesian lattice bhatanagar gross krook models we give the general solutions of lattices i e velocity sets and weights for the lattice bhatanagar gross krook lbgk models on two and three dimensional cartesian grids the solutions define the necessary and sufficient conditions so that the resulting lbgk model can accurately capture the dynamics of the moments retained in the distribution function in the parameter space of the weights the general solutions form low dimensional linear spaces from which minimal velocity sets are identified for the degrees of precision that are most relevant to the construction of high order lbgk models all well known lbgk lattices are found to be special cases of the given general solutions

a generalization of generalized quasi variational inequalities abstract the aim of this paper is to give an application of a fixed point theorem of a compact acyclic multifunction to a generalized quasi variational inequality problem in a non compact setting we present some general acyclic valued versions of several known results due to kim shih and tan

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

limit linear series basic theory in this paper we introduce techniques for handling the degeneration of linear series on smooth curves as the curves degenerate to a certain type of reducible curves curves of compact type the technically much simpler special case of 1 dimensional series was developed by beauville 2 knudsen 21 23 harris and mumford 17 in the guise of admissible covers it has proved very useful for studying the moduli space of curves the above papers and harris 16 and the simplest sorts of weierstrass points diaz 4 with our extended tools we are able to prove for example that in this paper we present the basic theory of limit linear series necessary for proving these results the results themselves will be taken up in our forthcoming papers 8 12 simpler applications not requiring the tools developed in this paper but perhaps clarified by them have already been given in our papers 5 7

a quadrature based third order moment method for dilute gas particle flows dilute gas particle flows can be described by a kinetic equation containing terms for spatial transport gravity fluid drag and particle particle collisions however the direct numerical solution of the kinetic equation is intractable for most applications due to the large number of independent variables a useful alternative is to reformulate the problem in terms of the moments of the velocity distribution function closure of the moment equations is challenging for flows away from the equilibrium maxwellian limit in this work a quadrature based third order moment closure is derived that can be applied to gas particle flows at any knudsen number a key component of quadrature based closures is the moment inversion algorithm used to find the weights and abscissas a robust inversion procedure is proposed for moments up to third order and tested for three example applications riemann shock problem impinging jets and vertical channel flow extension of the moment inversion algorithm to fifth or higher order is possible but left to future work the spatial fluxes in the moment equations are treated using a kinetic description and hence a gradient diffusion model is not used to close the fluxes because the quadrature based moment method employs the moment transport equations directly instead of a discretized form of the boltzmann equation the mass momentum and energy are conserved for arbitrary knudsen number including the euler limit while developed here for dilute gas particle flows quadrature based moment methods can in principle be applied to any application that can be modeled by a kinetic equation e g thermal and non isothermal flows currently treated using lattice boltzmann methods and examples are given from the literature

high order lattice boltzmann models for wall bounded flows at finite knudsen numbers we analyze a large number of high order discrete velocity models for solving the boltzmann bhatnagar gross krook equation for finite knudsen number flows using the chapman enskog formalism we prove for isothermal flows a relation identifying the resolved flow regimes for low mach numbers although high order lattice boltzmann models recover flow regimes beyond the navier stokes level we observe for several models significant deviations from reference results we found this to be caused by their inability to recover the maxwell boundary condition exactly by using supplementary conditions for the gas surface interaction it is shown how to systematically generate discrete velocity models of any order with the inherent ability to fulfill the diffuse maxwell boundary condition accurately both high order quadratures and an exact representation of the boundary condition turn out to be crucial for achieving reliable results for poiseuille flow we can reproduce the mass flow and slip velocity up to the knudsen number of 1 moreover for small knudsen numbers the knudsen layer behavior is recovered

lattice boltzmann models based on half range gauss hermite quadratures we discuss general features of thermal lattice boltzmann models based on half range gauss quadratures specialising to the half range gauss hermite and gauss laguerre cases the main focus of the paper is on the construction of high order half range hermite lattice boltzmann hhlb models the performance of the hhlb models is compared with that of laguerre lattice boltzmann llb and full range hermite lattice boltzmann hlb models by conducting convergence tests with respect to the quadrature order on stationary profiles of the particle number density macroscopic velocity temperature and heat fluxes in the two dimensional couette flow the bhatnagar gross krook bgk collision term is used throughout the paper to reduce the computational costs of the numerical simulations we use mixed lattice boltzmann models constructed using different quadrature methods on each cartesian axis for kn 0 01 the hlb models require the least number of velocities to satisfy our convergence test when kn 0 05 the hlb models are outperformed in terms of number of velocities employed by both the llb and the hhlb models moreover we find that the hhlb models require less quadrature points than the llb models at all tested values of kn which we attribute to the maxwellian form of the weight function for the half range hermite polynomials

on the three dimensional central moment lattice boltzmann method a three dimensional 3d lattice boltzmann method based on central moments is derived two main elements are the local attractors in the collision term and the source terms representing the effect of external and or self consistent internal forces for suitable choices of the orthogonal moment basis for the three dimensional twenty seven velocity d3q27 and its subset fifteen velocity d3q15 lattice models attractors are expressed in terms of factorization of lower order moments as suggested in an earlier work the corresponding source terms are specified to correctly influence lower order hydrodynamic fields while avoiding aliasing effects for higher order moments these are achieved by successively matching the corresponding continuous and discrete central moments at various orders with the final expressions written in terms of raw moments via a transformation based on the binomial theorem furthermore to alleviate the discrete effects with the source terms they are treated to be temporally semi implicit and second order with the implicitness subsequently removed by means of a transformation as a result the approach is frame invariant by construction and its emergent dynamics describing fully 3d fluid motion in the presence of force fields is galilean invariant numerical experiments for a set of benchmark problems demonstrate its accuracy

galilean invariant lattice boltzmann models with h theorem we demonstrate that the requirement of galilean invariance determines the choice of h function for a wide class of entropic lattice boltzmann models for the incompressible navier stokes equations the required h function has the form of the burg entropy for d 2 and of a tsallis entropy with q 1 2 d for d 2 where d is the number of spatial dimensions we use this observation to construct a fully explicit unconditionally stable galilean invariant lattice boltzmann model for the incompressible navier stokes equations for which attainable reynolds number is limited only by grid resolution

fundamental conditions for n th order accurate lattice boltzmann models in this paper we theoretically prove a set of fundamental conditions pertaining to discrete velocity sets and corresponding weights these conditions provide sufficient conditions for a priori formulation of lattice boltzmann models that automatically admit correct hydrodynamic moments up to any given n th order

general approach to constructing models of the boltzmann equation abstract the problem of thermodynamic parameterization of an arbitrary approximation of reduced description is solved on the base of this solution a new class of model kinetic equations is constructed that gives a model extension of the chosen approximation to a kinetic model model equations describe two processes rapid relaxation to the chosen approximation along the planes of rapid motions and the slow motion caused by the chosen approximation the h theorem is proved for these models it is shown that the rapid process always leads to entropy growth and also a neighborhood of the approximation is determined inside which the slow process satisfies the h theorem kinetic models for grad moment approximations and for the tamm mott smith approximation are constructed explicitly in particular the problem of concordance of the es model with the h theorem is solved

short note isotropic discrete laplacian operators from lattice hydrodynamics we show that discrete schemes developed for lattice hydrodynamics provide an elegant and physically transparent way of deriving laplacians with isotropic discretisation error isotropy is guaranteed whenever the laplacian weights follow from the discrete maxwell boltzmann equilibrium since these are by construction isotropic on the lattice we also point out that stencils using as few as 15 points in three dimensions generate isotropic laplacians these computationally efficient laplacians can be used in cell dynamical and hybrid lattice boltzmann simulations in favor of popular anisotropic laplacians which make use of larger stencils the method can be extended to provide discretisations of higher order and for other differential operators such the gradient divergence and curl

lattice boltzmann equation linear stability analysis thermal and athermal models although several thermal lattice boltzmann models have been proposed this method has not yet been shown to be able to describe nonisothermal fully compressible flows in a satisfactory manner mostly due to the presence of important deviations from the advection diffusion macroscopic equations and also due to numerical instabilities in this context this paper presents a linear stability analysis for some lattice boltzmann models that were recently derived as discrete forms of the continuous boltzmann equation p c philippi l a hegele jr l o e dos santos and r surmas phys rev e 63 056702 2006 in order to investigate the sources of instability and to find for each model the upper and lower limits for the macroscopic variables between which it is possible to ensure a stable behavior the results for two dimensional 2d lattices with 9 17 25 and 37 velocities indicate that increasing the order of approximation of the lattice boltzmann equation enhances stability results are also presented for an athermal 2d nine velocity model the accuracy of which has been improved with respect to the standard d2q9 model by adding third order terms in the lattice boltzmann equation

arbitrary lagrangian eulerian unstructured finite volume lattice boltzmann method for computing two dimensional compressible inviscid flows over moving bodies the objective of this study is to develop and apply an arbitrary lagrangian eulerian unstructured finite volume lattice boltzmann method ale fvlbm for solving two dimensional compressible inviscid flows around moving bodies the two dimensional compressible form of the lb equation is considered and the resulting lb equation is formulated in the ale framework on an unstructured body fitted mesh to correctly model the body shape and properly incorporate the mesh movement due to the body motion the spatial discretization of the resulting system of equations is performed by a second order cell centered finite volume method on arbitrary quadrilateral meshes and an implicit dual time stepping method is utilized for the time integration to stabilize the numerical solution appropriate numerical dissipation terms are added to the formulation at first the shock tube problem is computed to examine the accuracy of the solution obtained by applying the proposed fvlbm for this unsteady test case which includes shock expansion wave and contact discontinuity in the flow domain then the stationary isentropic vortex is simulated on both the stationary and moving meshes to assess the implementation of the geometric conservation law in enhancing the solution accuracy of the ale fvlbm the compressible inviscid flow in the transonic regime is then computed around the stationary naca0012 airfoil in order to further study the sensitivity of the solution method to the user defined parameters now the transonic inviscid flow is simulated over the pitching or plunging naca0012 airfoil to investigate the accuracy and capability of the proposed solution method ale fvlbm for the computation of the compressible flows over moving bodies finally the pitching or plunging naca0012 airfoil near the ground in the transonic inviscid flow is simulated as a practical and challenging problem to study the ground effect on the aerodynamic characteristics of the airfoil it is indicated that the solution methodology proposed based on the finite volume lbm formulated in the arbitrary lagrangian eulerian framework ale fvlbm is capable of accurately computing the compressible inviscid flows around the moving bodies with and without the ground effect

consistent lattice boltzmann schemes for the brinkman model of porous flow and infinite chapman enskog expansion we show that a consistent modeling of porous flows needs at least one free collision relaxation rate to avoid a nonlinear dependency of the numerical errors on the viscosity this condition is necessary to get the viscosity independent permeability from the stokes flow and to parametrize properly with nondimensional physical numbers the lattice boltzmann brinkman schemes the two relaxation time trt collision operator controls all coefficients of the higher order corrections in steady solutions with a specific combination of its two collision rates a possibility lacking for the bhatnagar gross krook bgk based single relaxation time schemes the analysis is based on exact recurrence equations of the evolution equation and illustrated for the exact solutions of the brinkman scheme in simply oriented parallel and diagonal channels the apparent viscosity coefficient of the trt stokes brinkman scheme in arbitrary flow is only approximated the compatibility of one dimensional arbitrarily rotated flows with the nonlinear navier stokes equilibrium is examined an explicit dependency for all coefficients on the relaxation rates is presented for the infinite steady state chapman enskog expansion

existence of solutions for lower semicontinuous quasi equilibrium problems in this paper we deal with the following quasi equilibrium problem given a nonempty subset c of a topological vector space x a nonempty set d two functions t c d f x d r and a multifunction s c 2x find x x such that x c x s x and f x tx f x tx for all x s x such a problem recently introduced in 1 extends and unifies several variational inequality problems generalized complementarity problems and nash equilibrium problems we prove several existence results in which the upper semicontinuity of the multifunction s is not assumed

asymptotic analysis of the lattice boltzmann equation in this article we analyze the lattice boltzmann equation lbe by using the asymptotic expansion technique we first relate the lbe to the finite discrete velocity model fdvm of the boltzmann equation with the diffusive scaling the analysis of this model directly leads to the incompressible navier stokes equations as opposed to the compressible navier stokes equations obtained by the chapman enskog analysis with convective scaling we also apply the asymptotic analysis directly to the fully discrete lbe as opposed to the usual practice of analyzing a continuous equation obtained through the taylor expansion of the lbe this leads to a consistency analysis which provides order by order information about the numerical solution of the lbe the asymptotic technique enables us to analyze the structure of the leading order errors and the accuracy of numerically derived quantities such as vorticity it also justifies the use of richardson s extrapolation method as an example a two dimensional taylor vortex flow is used to validate our analysis the numerical results agree very well with our analytic predictions

thermodynamic consistency of liquid gas lattice boltzmann simulations lattice boltzmann simulations have been very successful in simulating liquid gas and other multiphase fluid systems however the underlying second order analysis of the equation of motion has long been known to be insufficient to consistently derive the fourth order terms that are necessary to represent an extended interface these same terms are also responsible for thermodynamic consistency i e to obtain a true equilibrium solution with both a constant chemical potential and a constant pressure in this article we present an equilibrium analysis of nonideal lattice boltzmann methods of sufficient order to identify those higher order terms that lead to a lack of thermodynamic consistency we then introduce a thermodynamically consistent forcing method

variational and generalized variational inequalities with discontinuous mappings abstract in this paper we derive several existence results for the variational and the generalized variational inequality problems where the mappings under consideration need not have any continuity properties uniqueness of the solution of the generalized variational inequality problem is also investigated as a by product we obtain several existence and uniqueness results for the generalized complementarity problem applications to minimization problems with generalized convex functions are considered

optimizing the accuracy of lattice monte carlo algorithms for simulating diffusion the behavior of a lattice monte carlo lmc algorithm if it is designed correctly must approach that of the continuum system that it is designed to simulate as the time step and the mesh step tend to zero however we show for an algorithm for unbiased particle diffusion that if one of these two parameters remains fixed the accuracy of the algorithm is optimal for a certain finite value of the other parameter in one dimension the optimal algorithm with moves to the two nearest neighbor sites reproduces the correct second and fourth moments and minimizes the error for the higher moments at large times of the particle distribution and preserves the first two moments of the first passage time distributions in two and three dimensions the same level of accuracy requires simultaneous moves along two axes diagonal moves such moves attempting to cross an impenetrable boundary should be projected along the boundary rather than simply rejected we also treat the case of absorbing boundaries we discuss the relation between optimally accurate lmc algorithms and a particular case of lattice boltzmann lb algorithms for simulating diffusion and compare the computational efficiency of optimal lmc and optimal lb algorithms

on the generalized quasi variational inequality problems the purpose of this paper is to study the existence problem of solutions for some kinds of abstract generalized quasi variational inequality problems by using a new kind of fixed point approach

rotational invariance in the three dimensional lattice boltzmann method is dependent on the choice of lattice the lattice boltzmann method has recently gained popularity as a tool for simulating complex fluid flows it uses discrete sets of velocity vectors or lattices to create a reduced model of the molecular dynamics of a continuum fluid while several lattices are believed to behave isotropically there are reports of qualitatively incorrect results however thus far the reason as to why a lack of isotropy occurs is not known based on the hypothesis that lower order lattices may not display rotational invariance this study tests the isotropy of the d3q15 d3q19 and d3q27 lattices by performing simulations at intermediate reynolds numbers 50 500 and low knudsen number 250 the d3q15 and d3q19 lattices produced different results depending on the plane of the lattice with which the flow was aligned lattice planes with fewer than six velocity vectors consistently produced results which were qualitatively different from the planes with six or more velocity vectors these errors were not observed at re 50 or when a d3q27 lattice was used they appeared to be independent of grid density collision operator and ma this suggests that the lattices which contain these planes are not fully isotropic and therefore do not properly replicate the behavior of a real fluid in this particular situation notably downstream from the expansion predictions made using these models in more complex geometries may therefore be affected by the orientation of the lattice when using lbm in cfd simulation including validation this study highlights the need for caution to ensure that the solution obtained is independent of the lattice orientation throughout the domain

moment closure hierarchies for kinetic theories this paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory the first member of the hierarchy is the euler system which is based on maxwellian velocity distributions while the second member is based on nonisotropic gaussian velocity distributions the closure proceeds in two steps the first ensures that every member of the hierarchy is hyperbolic has an entropy and formally recovers the euler limit the second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct navier stokes behavior this is achieved through the introduction of a generalization of the bgk collision operator the simplest such system in three spatial dimensions is a 14 moment closure which also recovers the behavior of the grad 13 moment system when the velocity distributions lie near local maxwellians the closure procedure can be applied to a general class of kinetic theories

diffuse reflection boundary condition for high order lattice boltzmann models with streaming collision mechanism for lattice boltzmann lb models their particle feature is ensured by the streaming collision mechanism however this mechanism is often discarded for high order lb models due to multi speed lattices here we propose a new way of implementing the diffuse reflection boundary condition that can maintain the characteristic streaming collision mechanism for high order models it is then tested on four lattice models namely d2q16 d2q17 d2q37 and d3q121 for isothermal and thermal couette flows and lid driven cavity flows our implementation is shown to be able to achieve second order accuracy globally

thermal lattice boltzmann method based on a theoretically simple derivation of the taylor expansion we propose an approach to derive the thermal lattice boltzmann method that is based on the taylor expansion in variables of temperature as well as velocity and a direct calculation upon the gaussian quadrature based hypothesis this enables us to overcome the isothermal limitation and the low order accuracy simultaneously a systematic framework is explained for constructing numerically stable lattice boltzmann models the stability of this one dimensional lattice boltzmann model is demonstrated with a shock tube simulation

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

on the cubic velocity deviations in lattice boltzmann methods the macroscopic equations derived from the lattice boltzmann equation are not exactly the navier stokes equations here the cubic deviation terms and the methods proposed to eliminate them are studied the most popular two and three dimensional models d2q9 d3q15 d3q19 d3q27 are considered in the paper it is demonstrated that the compensation methods provide only partial elimination of the deviations for these models it is also shown that the compensation of qian and zhou 1998 europhys lett 42 359 using the compensation parameter k 1 in a d2q9 or d3q27 model can eliminate all the cross terms perfectly but the deviation terms x u3x y u3y and z u3z still survive the compensation

unified framework for a side by side comparison of different multicomponent algorithms lattice boltzmann vs phase field model lattice boltzmann models lbm and phase field models pfm are two of the most widespread approaches for the numerical study of multicomponent fluid systems both methods have been successfully employed by several authors but despite their popularity still remains unclear how to properly compare them and how they perform on the same problem here we present a unified framework for the direct one to one comparison of the multicomponent lbm against the pfm we provide analytical guidelines on how to compare the shan chen sc lattice boltzmann model for non ideal multicomponent fluids with a corresponding free energy fe lattice boltzmann model then in order to properly compare the lbm vs the pfm we propose a new formulation for the free energy of the cahn hilliard navier stokes equations finally the lbm model is numerically compared with the corresponding phase field model solved by means of a pseudo spectral algorithm this work constitute a first attempt to set the basis for a quantitative comparison between different algorithms for multicomponent fluids we limit our scope to the few of the most common variants of the two most widespread methodologies namely the lattice boltzmann model sc and fe variants and the phase field model

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

stable lattice boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves we follow the mathematical framework proposed by bouchut 2003 22 and present in this contribution a dual entropy approach for determining equilibrium states of a lattice boltzmann scheme this method is expressed in terms of the dual of the mathematical entropy relative to the underlying conservation law it appears as a good mathematical framework for establishing a h theorem for the system of equations with discrete velocities the dual entropy approach is used with d1q3 lattice boltzmann schemes for the burgers equation it conducts to the explicitation of three different equilibrium distributions of particles and induces naturally a nonlinear stability condition satisfactory numerical results for strong nonlinear shocks and rarefactions are presented we prove also that the dual entropy approach can be applied with a d1q3 lattice boltzmann scheme for systems of linear and nonlinear acoustics and we present a numerical result with strong nonlinear waves for nonlinear acoustics we establish also a negative result with the present framework the dual entropy approach cannot be used for the shallow water equations

lattice boltzmann model for high order nonlinear partial differential equations in this paper a general lattice boltzmann lb model is proposed for the high order nonlinear partial differential equation with the form ensuremath partial t ensuremath phi ensuremath sum k 1 m ensuremath alpha k ensuremath partial x k mathrm ensuremath pi k ensuremath phi 0 1 ensuremath le k ensuremath le m ensuremath le 6 ensuremath alpha k are constant coefficients mathrm ensuremath pi k ensuremath phi are some known differential functions of ensuremath phi as some special cases of the high order nonlinear partial differential equation the classical m kdv equation kdv burgers equation mathrm k n n burgers equation kuramoto sivashinsky equation and kawahara equation can be solved by the present lb model compared to the available lb models the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved in addition we also conducted a detailed chapman enskog analysis and found that the high order nonlinear partial differential equation can be correctly recovered from the proposed lb model finally a large number of simulations are performed and it is found that the numerical results agree with the analytical solutions and usually the present model is also more accurate than the existing lb models h lai and c ma sci china ser g 52 1053 2009 h lai and c ma phys a amsterdam 388 1405 2009 for high order nonlinear partial differential equations

a three dimensional lattice model for thermal compressible flow on standard lattices a three dimensional double distribution function thermal lattice boltzmann model has been developed for simulation of thermal compressible flows in the low mach number limit both the flow field and energy conservation equation are solved by lb approach a higher order density distribution function on standard lattices is used to solve the flow field while an energy distribution function is employed to compute the temperature field the equation of state of thermal perfect gas is recovered by higher order hermite polynomial expansions in navier stokes fourier equations the equilibrium distribution functions of d3q15 d3q19 and d3q27 lattices are obtained from the hermite expansion they exhibit slight differences originating in differences in the discrete lattice symmetries the correction terms in lb models for third order derivation are added using an external force in orthogonal polynomials form present models are successfully assessed considering several test cases namely the thermal couette flow rayleigh benard convection natural convection in square cavity and a spherical explosion in a 3d enclosed box the numerical results are in good agreement with both analytical solution and results given by previous authors

generalized quasi variational inequalities in locally convex topological vector spaces abstract let e be a hausdorff topological vector space and x e an arbitrary nonempty set denote by e the dual space of e and the pairing between e and e by w x for w e and x e given a point to set map s x 2x and a point to set map t x 2e the generalized quasi variational inequality problem gqvi is to find a point y s y and a point u t y such that re u y x 0 for all x s y by using the ky fan minimax principle or its generalized version as a tool some general theorems on solutions of the gqvi in locally convex hausdorff topological vector spaces are obtained which include a fixed point theorem due to ky fan and i l glicksberg and two different multivalued versions of the hartman stampacchia variational inequality

quasiequilibrium problem in h spaces in the present paper we study the quasiequilibrium problem and generalized quasiequilibrium problem of generalized quasi variational inequality in h spaces by a new method some new equilibrium existence theorems are given our results are different from corresponding given results or contain some recent results as their special cases c 2003 elsevier science ltd all rights reserved

from the continuous to the lattice boltzmann equation the discretization problem and thermal models the velocity discretization is a critical step in deriving the lattice boltzmann lbe from the continuous boltzmann equation this problem is considered in the present paper following an alternative approach and giving the minimal discrete velocity sets in accordance with the order of approximation that is required for the lbe with respect to the continuous boltzmann equation and with the lattice structure considering n to be the order of the polynomial approximation to the maxwell boltzmann equilibrium distribution it is shown that solving the discretization problem is equivalent to finding the inner product in the discrete space induced by the inner product in the continuous space that preserves the norm and the orthogonality of the hermite polynomial tensors in the hilbert space generated by the functions that map the velocity space onto the real numbers space as a consequence it is shown that for each order n of approximation the even parity velocity tensors are isotropic up to rank 2n in the discrete space the norm and the orthogonality restrictions lead to space filling lattices with increased dimensionality when compared with presently known lattices this problem is discussed in relation with a discretization approach based on a finite set of orthogonal functions in the discrete space two dimensional square lattices intended to be used in thermal problems and their respective equilibrium distributions are presented and discussed

the brill noether curve and prym tyurin varieties we prove that the jacobian of a general curve c of genus g 2a 1 with a ge 2 can be realized as a prym tyurin variety for the brill noether curve w 1 a 2 c as consequence of this result we are able to compute the class of the sum of secant divisors of the curve c embedded with a complete linear series g a 1 3a 2

theory of the lattice boltzmann equation symmetry properties of discrete velocity sets the lattice boltzmann equation replaces continuous particle velocity space by a finite set the velocity distribution function then varies over a finite dimensional vector space instead of over an infinite dimensional function space the number of linearly independent moments of the distribution function in a lattice boltzmann model cannot exceed the number of velocities finite dimensionality therefore necessarily induces linear dependences among the moments that do not exist in a continuous theory given a finite velocity set it is important to know which moments are free of these dependences elementary group theory is applied to the solution of this problem it is found that decomposing the velocity set into subsets that transform among themselves under an appropriate symmetry group makes it straightforward to uncover linear dependences among the moments the construction of some standard two and three dimensional models is reviewed from this viewpoint and procedures for constructing higher dimensional models are suggested

grad s approximation for moving and stationary walls in entropic lattice boltzmann simulations we introduce a new generalized wall boundary condition for the entropic lattice boltzmann method elbm that is capable of handling complex flow geometries including moving walls the grad s ten moments approximation is employed to approximate the missing populations at the boundary nodes the new boundary condition significantly improves accuracy stability and sub grid features of lbm and in particular of its entropic variant elbm the proposed boundary condition is tested for a range of flows in two dimensions including planar channel flow couette flow flow past a circular cylinder and sedimenting circular particles further simulations involving complex geometries such as flapping wing and transversely oscillating cylinder demonstrate superior stability and accuracy of the new wall boundary condition

generalized maxwell state and h theorem for computing fluid flows using the lattice boltzmann method generalized maxwell distribution function is derived analytically for the lattice boltzmann lb method all the previously introduced equilibria for lb are found as special cases of the generalized maxwellian the generalized maxwellian is used to derive a different class of multiple relaxation time lb models and prove the h theorem for them