stability analysis of lattice boltzmann methods the lattice boltzmann equation describes the evolution of the velocity distribution function on a lattice in a manner that macroscopic fluid dynamical behavior is recovered although the equation is a derivative of lattice gas automata it may be interpreted as a lagrangian finite difference method for the numerical simulation of the discrete velocity boltzmann equation that makes use of a bgk collision operator as a result it is not surprising that numerical instability of lattice boltzmann methods have been frequently encountered by researchers we present an analysis of the stability of perturbations of the particle populations linearized about equilibrium values corresponding to a constant density uniform mean flow the linear stability depends on the following parameters the distribution of the mass at a site between the different discrete speeds the bgk relaxation time the mean velocity and the wavenumber of the perturbations this parameter space is too large to compute the complete stability characteristics we report some stability results for a subset of the parameter space for a 7 velocity hexagonal lattice a 9 velocity square lattice and a 15 velocity cubic lattice results common to all three lattices are 1 the bgk relaxation time must be greater than 12 corresponding to positive shear viscosity 2 there exists a maximum stable mean velocity for fixed values of the other parameters and 3 as is increased from 12 the maximum stable velocity increases monotonically until some fixed velocity is reached which does not change for larger

a smooth transition model between kinetic and diffusion equations this paper presents a model which provides a smooth transition between a kinetic and a diffusion domain the idea is to use a buffer zone in which both diffusion and kinetic equations will be solved the solution of the original kinetic equation will be recovered as the sum of the solutions of these two equations we use an artificial connecting function which makes the equation on each domain degenerate at the end of the buffer zone thus no boundary condition is needed at the transition point this model avoids the delicate issue of finding the interface condition or iteration in a typical domain decomposition method that couples a kinetic equation with hydrodynamic equations a new asymptotic preserving method for this model is introduced and numerical examples are used to validate this new model and the new numerical method

operator theoretic differences between hardy and dirichlet type spaces abstract for 0 p the dirichlet type space d p 1 p consists of the analytic functions f in the unit disc d such that d f z p 1 z p 1 d a z motivated by operator theoretic differences between the hardy space h p and d p 1 p the integral operator t g f z 0 z f g d z d acting from one of these spaces to another is studied in particular it is shown on one hand that t g d p 1 p h p is bounded if and only if g bmoa when 0 p 2 and on the other hand that this equivalence is very far from being true if p 2 those symbols g such that t g d p 1 p h q is bounded or compact when p q are also characterized moreover the best known sufficient l type condition for a positive borel measure on d to be a p carleson measure for d p 1 p p 2 is significantly relaxed and the established result is shown to be sharp in a very strong sense

optimal low dispersion low dissipation lbm schemes for computational aeroacoustics lattice boltmzann methods lbm have been proved to be very effective methods for computational aeroacoustics caa which have been used to capture the dynamics of weak acoustic fluctuations in this paper we propose a strategy to reduce the dispersive and disspative errors of the two dimensional 2d multi relaxation time lattice boltzmann method mrt lbm by presenting an effective algorithm we obtain a uniform form of the linearized navier stokes equations corresponding to the mrt lbm in wave number space using the matrix perturbation theory and the equivalent modified equation approach for finite difference methods we propose a class of minimization problems to optimize the free parameters in the mrt lbm we obtain this way a dispersion relation preserving lbm drp lbm to circumvent the minimized dispersion error of the mrt lbm the dissipation relation precision is also improved and the stability of the mrt lbm with the small bulk viscosity is guaranteed von neuman analysis of the linearized mrt lbm is performed to validate the optimized dispersion dissipation relations considering monochromatic wave solutions meanwhile dispersion and dissipation errors of the optimized mrt lbm are quantitatively compared with the original mrt lbm finally some numerical simulations are carried out to assess the new optimized mrt lbm schemes

carleson measures and multipliers for dirichlet spaces abstract we characterize carleson measures on the dirichlet spaces our result leads to necessary and sufficient conditions for multipliers of the dirichlet spaces

evaluation of the external force term in the discrete boltzmann equation a representation of the forcing term in the boltzmann equation based on a hermite expansion of the boltzmann distribution function in velocity phase space is derived based on this description of the forcing term a systematic comparison of previous lattice boltzmann models describing a nonideal gas behavior is given

lattice boltzmann scheme for mixture modeling analysis of the continuum diffusion regimes recovering maxwell stefan model and incompressible navier stokes equations a finite difference lattice boltzmann scheme for homogeneous mixture modeling which recovers maxwell stefan diffusion model in the continuum limit without the restriction of the mixture averaged diffusion approximation was recently proposed p asinari phys rev e 77 056706 2008 the theoretical basis is the bhatnagar gross krook type kinetic model for gas mixtures p andries k aoki and b perthame j stat phys 106 993 2002 in the present paper the recovered macroscopic equations in the continuum limit are systematically investigated by varying the ratio between the characteristic diffusion speed and the characteristic barycentric speed it comes out that the diffusion speed must be at least one order of magnitude in terms of knudsen number smaller than the barycentric speed in order to recover the navier stokes equations for mixtures in the incompressible limit some further numerical tests are also reported in particular 1 the solvent and dilute test cases are considered because they are limiting cases in which the maxwell stefan model reduces automatically to fickian cases moreover 2 some tests based on the stefan diffusion tube are reported for proving the complete capabilities of the proposed scheme in solving maxwell stefan diffusion problems the proposed scheme agrees well with the expected theoretical results

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

near wall treatment for the simulation of turbulent flow by the cumulant lattice boltzmann method abstract we present a new wall function implementation for the cumulant lattice boltzmann method that sets a partial slip velocity on the wall by computing a skin frictional coefficient our approach uses local information and is particularly appropriate for implementations on general purpose graphics processing units the validation of the model has been conducted by performing numerical simulations of the turbulent channel flow test case with different grid resolutions and for different reynolds numbers the results showed encouraging agreement with direct numerical simulation data for velocity profile reynolds shear and normal stresses

lattice boltzmann simulations on irregular grids introduction of the natrium library the lattice boltzmann method is a modern approach to simulate fluid flow in its original formulation it is restricted to regular grids second order discretizations and a unity cfl number this paper describes our new off lattice boltzmann solver natrium an extensible and parallel c code to perform lattice boltzmann simulations on irregular grids natrium also allows high order spatial discretizations and non unity cfl numbers to be used we demonstrate how these features can efficiently decrease the number of grid points required in a simulation and thus reduce the computational time compared to the standard lattice boltzmann method we detail the implementation of a recently proposed semi lagrangian lattice boltzmann method and prove its efficiency in comparisons to other state of the art off lattice boltzmann schemes

lattice boltzmann model for the incompressible navier stokes equation in this paper a lattice boltzmann lb model to simulate incompressible flow is developed the main idea is to explicitly eliminate the terms of o m2 where m is the mach number due to the density fluctuation in the existing lb models in the proposed incompressible lb model the pressure p instead of the mass density is the independent dynamic variable the incompressible navier stokes equations are derived from the incompressible lb model via chapman enskog procedure numerical results of simulations of the plane poiseuille flow driven either by pressure gradient or a fixed velocity profile at entrance as well as of the 2d womersley flow are presented the numerical results are found to be in excellent agreement with theory

the navier stokes limit of the boltzmann equation for bounded collision kernels the present work establishes a navier stokes limit for the boltzmann equation considered over the infinite spatial domain r3 appropriately scaled families of diperna lions renormalized solutions are shown to have fluctuations whose limit points in the w l1 topology are governed by leray solutions of the limiting navier stokes equations this completes the arguments in bardos golse levermore commun pure appl math 46 5 667 753 1993 for the steady case and in lions masmoudi arch ration mech anal 158 3 173 193 2001 for the time dependent case

superposition operators hardy spaces and dirichlet type spaces abstract for 0 p and 1 the space of dirichlet type d p consists of those functions f which are analytic in the unit disc d and satisfy d 1 z f z p d a z the space d p 1 p is the closest one to the hardy space h p among all the d p our main object in this paper is studying similarities and differences between the spaces h p and d p 1 p 0 p regarding superposition operators namely for 0 p and 0 s we characterize the entire functions such that the superposition operator s with symbol maps the conformally invariant space q s into the space d p 1 p and also those which map d p 1 p into q s and we compare these results with the corresponding ones with h p in the place of d p 1 p we also study the more general question of characterizing the superposition operators mapping d p into q s and q s into d p for any admissible triplet of numbers p s

finite difference lattice boltzmann method for compressible thermal fluids a splitting method to solve the modeled lattice boltzmann equation the splitting technique permits the boundary conditions for the lattice boltzmann equation to be set as conveniently as those required for the finite difference solution of the navier stokes equations it is shown that the compressible navier stokes equation can be recovered fully from this approach however the formulation requires the solution of a poisson equation governing a secondorder tensor thus constructed the method has no arbitrary constants the proposed method is used to simulate thermalcouette flow aeroacoustics andshockstructureswithanextendedthermodynamicsmodel thesimulations are carried out using a high order finite difference scheme with a two dimensional nine velocity lattice all simulations are performed using a single relaxation time and a set of constants deduced from the derivation it is found that the finite difference lattice boltzmann method is able to correctly replicate viscous effects in thermal couette flows aeroacoustics andshockstructures thesolutionsobtainedareidenticaleithertoanalyticalresults or obtained by solving the compressible navier stokes equations using a direct numerical simulation technique

differential evolution with preferential crossover we study the mutation operation of the differential evolution algorithm in particular we study the effect of the scaling parameter of the differential vector in mutation we derive the probability density function of points generated by mutation and thereby identify some drawbacks of the scaling parameter we also visualize the drawbacks using simulation we then propose a crossover rule called the preferential crossover rule to reduce the drawbacks the preferential crossover rule uses points from an auxiliary population set we also introduce a variable scaling parameter in mutation motivations for these changes are provided a numerical study is carried out using 50 test problems many of which are inspired by practical applications numerical results suggest that the proposed modification reduces the number of function evaluations and cpu time considerably

from the boltzmann equations to the equations of incompressible fluid mechanics i we consider here the problem of deriving rigorously from boltzmann s equation globally in time and for general initial conditions fluid mechanics equations such as the navier stokes or euler equations

lattice boltzmann model for the compressible navier stokes equations with flexible specific heat ratio we have developed a lattice boltzmann model for the compressible navier stokes equations with a flexible specific heat ratio several numerical results are presented and they agree well with the corresponding solutions of the navier stokes equations in addition an explicit finite difference scheme is proposed for the numerical calculation that can make a stable calculation with a large courant number

adiabatic perturbations for compactons under dissipation and numerically induced dissipation compacton propagation under dissipation shows amplitude damping and the generation of tails the numerical simulation of compactons by means of dissipative schemes also show the same behaviors the truncation error terms of a numerical method can be considered as a perturbation of the original partial differential equation and perturbation methods can be applied to its analysis for dissipative schemes or when artificial dissipation is added the adiabatic perturbation method yields evolution equations for the amplitude loss in the numerical solution and the amplitude of the numerically induced tails in this paper such methods are applied to the k 2 2 rosenau hyman equation showing a very good agreement between perturbative and numerical results

a critical analysis of the modified equation technique of warming and hyett abstract warming and hyett developed a modified equation technique in which the behavior of a difference scheme is evaluated by using the coefficients of a certain modified equation specifically they discovered a connection between these coefficients and the multiplication factor obtained from the von neumann analysis since the dissipation and dispersion off error components are determined by the multiplication factor the former properties can be studied using the coefficients of the modified equation the work of warming and hyett represents a key step in the development of the method of modified equations through this work it became clear that modified equations should be derived from the difference scheme rather than from the original differential equation however in order to prove the above connection warming and hyett incorrectly interpreted their modified equations as the actual partial differential equations solved by the difference schemes the main purpose of the current study is to investigate rigorously the above connection without using their interpretation the result of this investigation shows that the above connection is only partially valid for multilevel schemes in the von neumann analysis the multiplication factor associated with a wave number generally has l 1 roots for an l level scheme it is shown that the coefficients of the modified equation provide information for only the principal root

straight velocity boundaries in the lattice boltzmann method various ways of implementing boundary conditions for the numerical solution of the navier stokes equations by a lattice boltzmann method are discussed five commonly adopted approaches are reviewed analyzed and compared including local and nonlocal methods the discussion is restricted to velocity dirichlet boundary conditions and to straight on lattice boundaries which are aligned with the horizontal and vertical lattice directions the boundary conditions are first inspected analytically by applying systematically the results of a multiscale analysis to boundary nodes this procedure makes it possible to compare boundary conditions on an equal footing although they were originally derived from very different principles it is concluded that all five boundary conditions exhibit second order accuracy consistent with the accuracy of the lattice boltzmann method the five methods are then compared numerically for accuracy and stability through benchmarks of two dimensional and three dimensional flows none of the methods is found to be throughout superior to the others instead the choice of a best boundary condition depends on the flow geometry and on the desired trade off between accuracy and stability from the findings of the benchmarks the boundary conditions can be classified into two major groups the first group comprehends boundary conditions that preserve the information streaming from the bulk into boundary nodes and complete the missing information through closure relations boundary conditions in this group are found to be exceptionally accurate at low reynolds number boundary conditions of the second group replace all variables on boundary nodes by new values they exhibit generally much better numerical stability and are therefore dedicated for use in high reynolds number flows

on a superconvergent lattice boltzmann boundary scheme in a seminal paper 20 ginzburg and adler 1994 analyzed the bounce back boundary conditions for the lattice boltzmann scheme and showed that it could be made exact to second order for the poiseuille flow if some expressions depending upon the parameters of the method were satisfied thus defining so called magic parameters using the taylor expansion method that one of us developed we analyze a series of simple situations 1d and 2d for diffusion and for linear fluid problems using bounce back and anti bounce back numerical boundary conditions the result is that magic parameters depend upon the detailed choice of the moments and of their equilibrium values they may also depend upon the way the flow is driven

analysis of lattice boltzmann initialization routines lb simulations can be affected by the arising of initial layers due to an inconsistent initialization of the discrete lb populations we present some previously proposed initialization routines built to overcome that problem using the asymptotic expansion technique we show how their features can be analyzed and in some cases how accuracy and computational efficiency can be improved

comparison between lattice boltzmann method and navier stokes high order schemes for computational aeroacoustics computational aeroacoustic caa simulation requires accurate schemes to capture the dynamics of acoustic fluctuations which are weak compared with aerodynamic ones in this paper two kinds of schemes are studied and compared the classical approach based on high order schemes for navier stokes like equations and the lattice boltzmann method the reference macroscopic equations are the 3d isothermal and compressible navier stokes equations a von neumann analysis of these linearized equations is carried out to obtain exact plane wave solutions three physical modes are recovered and the corresponding theoretical dispersion relations are obtained then the same analysis is made on the space and time discretization of the navier stokes equations with the classical high order schemes to quantify the influence of both space and time discretization on the exact solutions different orders of discretization are considered with and without a uniform mean flow three different lattice boltzmann models are then presented and studied with the von neumann analysis the theoretical dispersion relations of these models are obtained and the error terms of the model are identified and studied it is shown that the dispersion error in the lattice boltzmann models is only due to the space and time discretization and that the continuous discrete velocity boltzmann equation yield the same exact dispersion as the navier stokes equations finally dispersion and dissipation errors of the different kind of schemes are quantitatively compared it is found that the lattice boltzmann method is less dissipative than high order schemes and less dispersive than a second order scheme in space with a 3 step runge kutta scheme in time the number of floating point operations at a given error level associated with these two kinds of schemes are then compared

the acoustic limit for the boltzmann equation the acoustic equations are the linearization of the compressible euler equations about a spatially homogeneous fluid state we first derive them directly from the boltzmann equation as the formal limit of moment equations for an appropriately scaled family of boltzmann solutions we then establish this limit for the boltzmann equation considered over a periodic spatial domain for bounded collision kernels appropriately scaled families of diperna lions renormalized solutions are shown to have fluctuations that converge entropically and hence strongly in l 1 to a unique limit governed by a solution of the acoustic equations for all time provided that its initial fluctuations converge entropically to an appropriate limit associated to any given l 2 initial data of the acoustic equations the associated local conservation laws are recovered in the limit

a knudsen layer theory for lattice gases abstract a knudsen layer theory is presented for lattice gases with arbitrary boundary conditions analytical results are obtained for two special orientations these exhibit anisotropic knudsen layers provided suitable conditions are satisfied however the standard boundary conditions used in previous simulations are shown to be isotropic the bulk steady state extending everywhere in the gas this theory allows a more accurate localization of the obstacle with respect to the lattice nodes these results are in good agreement with the numerical simulations

a lattice boltzmann method for immiscible two phase stokes flow with a local collision operator we propose a lattice boltzmann model for immiscible two phase stokes flow with a local collision operator the model is based on two different lattice boltzmann automata one for the flow field and one for an indicator function for the two different phases the model is described in detail and verified by the following test cases a static bubble for the surface tension a closed capillary tube for the contact angle and two phase flow in a concentric annulus for the viscosity ratio in the appendix an asymptotic analysis for the derivation of the two phase stokes equation is given

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

extensive analysis of the lattice boltzmann method on shifted stencils standard lattice boltzmann methods lbms are based on a symmetric discretization of the phase space which amounts to study the evolution of particle distribution functions pdfs in a reference frame at rest this choice induces a number of limitations when the simulated flow speed gets closer to the sound speed such as velocity dependent transport coefficients the latter issue is usually referred to as a galilean invariance defect to restore the galilean invariance of lbms it was proposed to study the evolution of pdfs in a comoving reference frame by relying on asymmetric shifted lattices n frapolli s s chikatamarla and i v karlin phys rev lett 117 010604 2016 from the numerical viewpoint this corresponds to overcoming the rather restrictive courant friedrichs lewy conditions on standard lbms and modeling compressible flows while keeping memory consumption and processing costs to a minimum therefore using the standard first neighbor stencils in the present work systematic physical error evaluations and stability analyses are conducted for different discrete equilibrium distribution functions edfs and collision models thanks to them it is possible to 1 better understand the effect of this solution on both physics and stability 2 assess its viability as a way to extend the validity range of lbms and 3 quantify the importance of the reference state as compared to other parameters such as the equilibrium state and equilibration path the results clearly show that in theory the concept of shifted lattices allows the scheme to deal with arbitrarily high values of the nondimensional velocity furthermore just like the zero mach flow for the standard stencils it is observed that setting the shift velocity to the fluid velocity results in optimal physical and numerical properties in addition a detailed analysis of the obtained results shows that the properties of different collision models and edfs remain unchanged under the shift of stencil in other words by introducing a velocity shift in the stencil the optimal operating point in terms of physics and numerics will also be shifted by the same vector regardless of the edf or collision model considered eventually while limited to the d2q9 stencil with the nine possible first neighbor shifts the present study and corresponding conclusions can be extended to other stencils and velocity shifts in a straightforward manner

discrete adjoint sensitivity analysis for fluid flow topology optimization based on the generalized lattice boltzmann method abstract a discrete adjoint sensitivity analysis for fluid flow topology optimization based on the lattice boltzmann method lbm with multiple relaxation times mrt is developed the lattice boltzmann fluid solver is supplemented by a porosity model using a darcy force the continuous transition from fluid to solid facilitates a gradient based optimization process of the design topology of fluidic channels the adjoint lbm equation which is used to compute the gradient of the optimization objective with respect to the design variables is derived in moment space and found to be as simple as the original lbm the moment based spatial momentum derivatives used to express the discrete objective functional cost function have the advantage that the local stress tensor is a local quantity avoiding the numerical computation of gradients of the discrete velocity field this is particularly useful if dissipation is a design criterion as demonstrated in this paper the method is validated by a detailed comparison with results obtained by borrvall et al for stokes flow while their approach is only valid for stokes flow i e very low reynolds numbers our approach in its present form can in principle be applied for flows of different reynolds numbers just like the navier stokes equation based approaches this point is demonstrated with a bending pipe example for various reynolds numbers

computation of gas solid flows by finite difference boltzmann equation abstract in this paper we will discuss in detail how to use a finite difference lattice boltzmann equation model in which an external force term is involved to simulate two way coupling gas solid flows the numerical results are found to be in good agreement with analytical data and some other numerical results

lattice boltzmann method for the compressible euler equations the lattice boltzmann model for the compressible euler equations is proposed together with its rigorous theoretical background the proposed model has completely overcome the defects of the previous model that the specific heat ratio cannot be chosen freely the macroscopic variables obtained from the solution are shown to satisfy in the limit of the small knudsen number the compressible euler equations if the variation of the solution is moderate this is the case where no shock waves or contact discontinuities appear in contrast when the solution makes steep variation at several localized regions due to the appearance of shock waves and contact discontinuities the corresponding macroscopic variables satisfy the weak form of the euler equations their derivation is carried out rigorously by taking into account the scale of variation of the solution correctly this is the first study that has laid the theoretical foundation of the lattice boltzmann model for the simulation of flows with shock waves and contact discontinuities numerical examples and the error estimates are also given which are consistent with the above theoretical arguments

computation of flow noise using source terms in linearized euler s equations an acoustic analogy using linearized euler s equations lee forced with aerodynamic source terms is investigated to computetheacousticfare eld thishybridmethod isappliedto threemodelproblemssimulatedby solving navier stokes equations in this way its validity is estimated by comparing the predicted acoustic e eld with the reference solution given directly by the navier stokes equations the noise radiated by two corotating vortices is studied e rst in a medium at rest and second in a mean sheared e ow with no convection velocity then the sound e eld generated by vortex pairings in a subsonic mixing layer is investigated in this case a simplie ed formulation of lee is proposed to prevent the exponential growth of instability waves the acoustic e elds obtained by solving lee are in good agreement with the reference solution this study shows that the source terms introduced into the lee are appropriate for free sheared e ows and that acoustic mean e ow interactions are properly taken into account in the wave operator nomenclature b half width of the monopolar source c sound velocity e f h vectors in linearized euler s equations lee f frequency f0 fundamental frequency of the mixing layer k complex wave number kr ciki m mach number p pressure re reynolds number rc vortex core radius r0 initial half distance between the two vortices s sound source vector in lee si source terms in the momentum equations t period tij lighthill s tensor t time u unknown vector in lee u1 slow stream velocity of the mixing layer u2 rapid stream velocity of the mixing layer u velocity vector u1 u2 v initial tangential velocity of vortices

fixed grid fluid structure interaction in two dimensions based on a partitioned lattice boltzmann and p fem approach over the last decade the lattice boltzmann method which was derived from the kinetic gas theory has matured as an efficient approach for solving navier stokes equations the p fem approach has proved to be highly efficient for a variety of problems in the field of structural mechanics our goal is to investigate the validity and efficiency of coupling the two approaches to simulate transient bidirectional fluid structure interaction problems with geometrically non linear structural deflections a benchmark configuration of self induced large oscillations for a flag attached to a cylinder can be accurately and efficiently reproduced within this setting we describe in detail the force evaluation techniques displacement transfers and the algorithm used to couple these completely different solvers as well as the results and compare them with a benchmark reference solution computed by a monolithic finite element approach copyright 2009 john wiley sons ltd

asymptotic analysis of lattice boltzmann boundary conditions in this article we use a general method for the analysis of finite difference schemes to investigate lattice boltzmann algorithms for navier stokes problems with dirichlet boundary conditions several link based boundary conditions for commonly used lattice boltzmann bgk models are considered with our method the accuracy of the algorithms can be exactly predicted moreover the analytical results can be used to construct new algorithms which is demonstrated with a corrected bounce back rule that requires only local evaluations but still yields second order accuracy for the velocity the analysis is applicable to general geometries and instationary flows

composition operators acting on holomorphic sobolev spaces we study the action of composition operators on sobolev spaces of analytic functions having fractional derivatives in some weighted bergman space or hardy space on the unit disk criteria for when such operators are bounded or compact are given in particular we find the precise range of orders of fractional derivatives for which all composition operators are bounded on such spaces sharp results about boundedness and compactness of a composition operator are also given when the inducing map is polygonal

new mutation schemes for differential evolution algorithm and their application to the optimization of directional over current relay settings differential evolution is a novel evolutionary approach capable of handling non differentiable nonlinear and multimodal objective functions it has been consistently ranked as one of the best search algorithm for solving global optimization problems in several case studies in the present study we propose five new mutation schemes for the basic de algorithm the corresponding versions are termed as mde1 mde2 mde3 mde4 and mde5 these new schemes make use of the absolute weighted difference between the two points and instead of using a fixed scaling factor f use a scaling factor following the laplace distribution the performance of the proposed schemes is validated empirically on a suit of ten benchmark problems having box constraints numerical analysis of results shows that the proposed schemes improves the convergence rate of the de algorithm and also maintains the quality of solution efficiency of the proposed schemes is further validated by applying it to a real life electrical engineering problem dealing with the optimization of directional over current relay settings it is a highly constrained nonlinear optimization problem a constraint handling mechanism based on repair methods is used for handling the constraints once again the simulation results show the compatibility of the proposed schemes for solving the real life problem

a spectral difference lattice boltzmann method for solution of inviscid compressible flows on structured grids in this work a spectral difference lattice boltzmann method sdlbm is developed and applied for an accurate simulation of two dimensional inviscid compressible flows on structured grids the compressible form of the discrete boltzmann bgk equation is used in which multiple particle speeds have to be employed to correctly model the compressibility in a thermal fluid here the 2d compressible lattice boltzmann lb model proposed by watari is applied the spectral difference sd method is implemented for the solution of the lb equation in which the particle distribution function is stored at the solution points while the fluxes are computed at the flux points for calculating the particle distribution function for time accurate solutions the fourth order runge kutta scheme is used to discretize the temporal term in the lb equation note that the procedure of implementing the sd method to solve the lb equation is nearly the same as that procedure developed in the literature for the solution of the euler equations the accuracy and robustness of the present solution methodology are demonstrated by simulating different benchmark compressible flow problems a sensitivity study is also conducted to evaluate the effects of the numerical parameters and the grid size distribution on the accuracy and performance of the solution at first three inviscid compressible flow problems namely the stationary isentropic vortex the shock tube and the shock vortex interaction are solved by using the sdlbm to demonstrate the accuracy and robustness of the present solution methodology results computed for these test cases are in good agreement with the analytical and the available numerical solutions to more assess the accuracy and robustness of the sdlbm a third order finite volume lbm fvlbm is also developed and the solutions obtained by these two methods for the test cases simulated are thoroughly compared with each other it is demonstrated that the sdlbm is more accurate and robust compressible inviscid flow solver that can be used for evaluating the other compressible lb based flow solvers as the final problem the supersonic flow past a bump is simulated to demonstrate the accuracy and robustness of the sdlbm in reaching to a steady state solution

steady state convergence acceleration of the generalized lattice boltzmann equation with forcing term through preconditioning several applications exist in which lattice boltzmann methods lbm are used to compute stationary states of fluid motions particularly those driven or modulated by external forces standard lbm being explicit time marching in nature requires a long time to attain steady state convergence particularly at low mach numbers due to the disparity in characteristic speeds of propagation of different quantities in this paper we present a preconditioned generalized lattice boltzmann equation glbe with forcing term to accelerate steady state convergence to flows driven by external forces the use of multiple relaxation times in the glbe allows enhancement of the numerical stability particular focus is given in preconditioning external forces which can be spatially and temporally dependent in particular correct forms of moment projections of source forcing terms are derived such that they recover preconditioned navier stokes equations with non uniform external forces as an illustration we solve an extended system with a preconditioned lattice kinetic equation for magnetic induction field at low magnetic prandtl numbers which imposes lorentz forces on the flow of conducting fluids computational studies particularly in three dimensions for canonical problems show that the number of time steps needed to reach steady state is reduced by orders of magnitude with preconditioning in addition the preconditioning approach resulted in significantly improved stability characteristics when compared with the corresponding single relaxation time formulation

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

accuracy of non newtonian lattice boltzmann simulations this work deals with the accuracy of non newtonian lattice boltzmann simulations previous work for newtonian fluids indicate that depending on the numerical value of the dimensionless collision frequency additional artificial viscosity is introduced which negatively influences the accuracy since the non newtonian fluid behavior is incorporated through appropriate modeling of the dimensionless collision frequency a dependent error e is introduced and its influence on the overall error is investigated here simulations with the srt and the mrt model are carried out for power law fluids in order to numerically investigate the accuracy of non newtonian lattice boltzmann simulations a goal of this accuracy analysis is to derive a recommendation for an optimal choice of the time step size and the simulation mach number respectively for the non newtonian case an error estimate for e in the form of a functional is derived on the basis of a series expansion of the lattice boltzmann equation this functional can be solved analytically for the case of the hagen poiseuille channel flow of non newtonian fluids with the help of the error functional the prediction of the global error minimum of the velocity field is excellent in regions where the e error is the dominant source of error with an optimal simulation mach number the simulation is about one order of magnitude more accurate additionally for both collision models a detailed study of the convergence behavior of the method in the non newtonian case is conducted the results show that the simulation mach number has a major impact on the convergence rate and second order accuracy is not preserved for every choice of the simulation mach number

an onsager like relation for the lattice boltzmann method an onsager like relation is proposed as a new criterion for constructing and analyzing the lattice boltzmann lb method for lb models obeying the relation we analyze their linearized stability establish their diffusive limit and find new constraints for those with free parameters the new relation seems of fundamental importance for the lb method

nonexistence of h theorem for some lattice boltzmann models in this paper we provide a set of sufficient conditions under which a lattice boltzmann model does not admit an h theorem by verifying the conditions we prove that a number of existing lattice boltzmann models does not admit an h theorem these models include d2q6 d2q9 and d3q15 athermal models and d2q16 and d3q40 thermal energy conserving models the proof does not require the equilibria to be polynomials

lattice boltzmann hydrodynamics of single square grid generated turbulence a partial entropic stabilisation approach abstract under resolved direct numerical simulations of turbulence in the wake of a single square grid ssg are conducted using the lattice boltzmann method lbm with varying degrees of partial moment entropic stabilisation turbulence statistics along the grid centreline show satisfactory agreement with literature greatly outperforming a control study employing the single relaxation time smagorinsky les model with equivalent resolution the coarse resolution entropic lbm simulations are able to capture the qualitative behaviour of the anisotropic and non homogeneous turbulence generated by the ssg indicating possible extension of the capability of the entropic lattice boltzmann method to simulate more exotic forms of turbulence beyond those reported by the majority of current studies however in the extreme near field of the single square grid pseudo compressibility effects even at a low lattice mach number undermine lbm predictions for turbulence which is generated by action of the pressure hessian this is found to be an issue for all models studied and not a numerical artefact of the entropic stabilisation with regard to the action of the entropic stabiliser a comparison of models employing differing degrees of stabilisation on higher order natural moments suggest that the trace of the stress tensor should not be subject to entropic control for better accuracy in turbulence prediction

an interpretation and derivation of the lattice boltzmann method using strang splitting the lattice boltzmann space time discretisation as usually derived from integration along characteristics is shown to correspond to a strang splitting between decoupled streaming and collision steps strang splitting offers a second order accurate approximation to evolution under the combination of two non commuting operators here identified with the streaming and collision terms in the discrete boltzmann partial differential equation strang splitting achieves second order accuracy through a symmetric decomposition in which one operator is applied twice for half timesteps and the other operator is applied once for a full timestep we show that a natural definition of a half timestep of collisions leads to the same change of variables that was previously introduced using different reasoning to obtain a second order accurate and explicit scheme from an integration of the discrete boltzmann equation along characteristics this approach extends easily to include general matrix collision operators and also body forces finally we show that the validity of the lattice boltzmann discretisation for grid scale reynolds numbers larger than unity depends crucially on the use of a crank nicolson approximation to discretise the collision operator replacing this approximation with the readily available exact solution for collisions uncoupled from streaming leads to a scheme that becomes much too diffusive due to the splitting error unless the grid scale reynolds number remains well below unity

an interesting class of partial differential equations this paper presents an observation that under reasonable conditions many partial differential equations from mathematical physics possess three structural properties one of them can be understand as a variant of the celebrated onsager reciprocal relation in modern thermodynamics it displays a direct relation of irreversible processes to the entropy change we show that the properties imply various entropy dissipation conditions for hyperbolic relaxation problems as an application of the observation we propose an approximation method to solve relaxation problems moreover the observation is interpreted physically and verified with eight sets of systems from different fields

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

a self adaptive chaotic differential evolution algorithm using gamma distribution for unconstrained global optimization evolutionary algorithms eas have yielded promising results for solving nonlinear non differentiable and multi modal optimization problems due to its population based nature eas can avoid being trapped in a local optimum and consequently have the ability to find global optimal solutions as a novel evolutionary technique differential evolution de has received increasing attention and wide applications in a variety of fields de algorithm uses an efficient way of self adapting mutation using small populations for function optimization over continuous spaces the potentialities of de are its simple structure easy use convergence property quality of solution and robustness in this paper an effective self adaptive de algorithm based on gaussian probability distribution gamma distribution and chaotic sequence degc for solving continuous global optimization problems is proposed the proposed degc algorithm is tested on several benchmark functions from the usual literature numerical results comparisons with a classical de approach and a self adaptive de approach demonstrate the effectiveness and efficiency of the proposed degc algorithm

a consistent grid coupling method for lattice boltzmann schemes a method of coupling grids of different mesh size is developed for classical lattice boltzmann lb algorithms on uniform grids the approach is based on an asymptotic analysis revealing suitable quantities equalized along the grid interfaces for exchanging information between the subgrids in contrast to other couplings the method works without overlap zones moreover the grid velocity mach number is not kept constant as the time step depends not linearly but quadratically on the grid spacing to illustrate the basic idea we use a simple lb algorithm solving the advection diffusion equation the proposed grid coupling is validated by numerical convergence studies indicating that the coupling does not affect the second order convergence behavior of the lb algorithm which is observed on uniform grids in order to demonstrate its principal applicability to other lb models the coupling is generalized to the standard d2p9 model for navier stokes flow and tested numerically as we use analytic tools different from the chapman enskog expansion the theoretical background material is given in two appendices in particular the results of numerical experiments are confirmed with a consistency analysis

initial and boundary conditions for the lattice boltzmann method an alternative approach of implementing initial and boundary conditions for the lattice boltzmann method is presented the basic idea is to calculate the lattice boltzmann populations at a boundary node from the fluid variables that are specified at this node using the gradients of the fluid velocity the numerical performance of the lattice boltzmann method is tested on several problems with exact solutions and is also compared to an explicit finite difference projection method the discretization error of the lattice boltzmann method decreases quadratically with finer resolution both in space and in time the roundoff error of the lattice boltzmann method creates problems unless double precision arithmetic is used

nonexistence of h theorems for the athermal lattice boltzmann models with polynomial equilibria we prove that no h theorem exists for the athermal lattice boltzmann equation with polynomial equilibria satisfying the conservation laws exactly and explicitly the proof is demonstrated by using the seven velocity model in a triangular lattice in two dimensions and can be readily extended to other lattice boltzmann models in two and three dimensions some issues pertinent to the numerical instabilities of the lattice boltzmann method are disscussed

a stability notion for lattice boltzmann equations lattice boltzmann equations are usually constructed to satisfy some physical requirements such as galilean invariance and isotropy to possess a velocity independent pressure and no compressible effects and so on in this paper a stability requirement is introduced as a new criterion for the constructions with this requirement we derive some relations of parameters for several lattice boltzmann models interestingly these relations are satisfied by many choices of parameters used in the literature

weighted mathbb l 2 stability of the lattice boltzmann method this article is concerned with the linearized stability of the lattice boltzmann method both on periodic domains and on bounded domains with the bounce back rule used at the boundaries under a structural hypothesis we prove that a weighted mathbb l 2 norm of the solutions to the linearized lattice boltzmann method is decreasing with time moreover we show that the structural hypothesis holds true for many lattice boltzmann models

second order interpolation of the flow field in the lattice boltzmann method we propose a scheme to reconstruct the flow field up to second order in a cell composed of four lattices nodes in 2 dimensions the information contained in the higher order moments of the distribution functions is used to construct an interpolation scheme of second order for the velocity field

boundary forces in lattice boltzmann analysis of momentum exchange algorithm when lattice boltzmann methods are used to simulate fluid structure interaction problems they need to be coupled with additional routines to evaluate the boundary forces without destroying the efficiency and accuracy of the original method we use the asymptotic expansion technique to analyze one such approach the momentum exchange algorithm investigating its properties in detail whether it can be improved and in which cases it can be successfully used a statement regarding the accuracy is presented together with results of numerical tests which illustrate the theoretical considerations

dynamic subgrid scale modeling of turbulent flows using lattice boltzmann method abstract in this paper we discuss the incorporation of dynamic subgrid scale sgs models in the lattice boltzmann method lbm for large eddy simulation les of turbulent flows the use of a dynamic procedure which involves sampling or test filtering of super grid turbulence dynamics and subsequent use of scale invariance for two levels circumvents the need for empiricism in determining the magnitude of the model coefficient of the sgs models we employ the multiple relaxation times mrt formulation of lbm with a forcing term which has improved physical fidelity and numerical stability achieved by proper separation of relaxation time scales of hydrodynamic and non hydrodynamic modes for simulation of the grid filtered dynamics of large eddies the dynamic procedure is illustrated for use with the common smagorinsky eddy viscosity sgs model and incorporated in the lbm kinetic approach through effective relaxation time scales the strain rate tensor in the sgs model is locally computed by means of non equilibrium moments of the mrt lbm we also discuss proper sampling techniques or test filters that facilitate implementation of dynamic models in the lbm for accommodating variable resolutions we employ conservative locally refined grids in this framework as examples we consider the canonical anisotropic and inhomogeneous turbulent flow problem i e fully developed turbulent channel flow at two different shear reynolds numbers r e of 180 and 395 the approach is able to automatically and self consistently compute the values of the smagorinsky coefficient c s in particular the computed value in the outer or bulk flow region where turbulence is generally more isotropic is about 0 155 or the model coefficient c c s 2 0 024 which is in good agreement with prior data it is also shown that the model coefficient becomes smaller and approaches towards zero near walls reflecting the dampening of turbulent length scales near walls the computed turbulence statistics at these reynolds numbers are also in good agreement with prior data the paper also discusses a procedure for incorporation of more general scale similarity based sgs stress models

two dimensional lattice boltzmann model for compressible flows with high mach number abstract in this paper we present an improved lattice boltzmann model for compressible navier stokes system with high mach number the model is composed of three components i the discrete velocity model by m watari and m tsutahara phys rev e 67 2003 036306 ii a modified lax wendroff finite difference scheme where reasonable dissipation and dispersion are naturally included iii artificial viscosity the improved model is convenient to compromise the high accuracy and stability the included dispersion term can effectively reduce the numerical oscillation at discontinuity the added artificial viscosity helps the scheme to satisfy the von neumann stability condition shock tubes and shock reflections are used to validate the new scheme in our numerical tests the mach numbers are successfully increased up to 20 or higher the flexibility of the new model makes it suitable for tracking shock waves with high accuracy and for investigating nonlinear nonequilibrium complex systems

proteus a coupled iterative force correction immersed boundary multi domain cascaded lattice boltzmann solver abstract most realistic fluid flow problems are characterized by high reynolds numbers and complex boundaries over the last ten years immersed boundary methods that are able to cope with realistic geometries have been applied to lattice boltzmann lb methods these methods however have normally been applied to low reynolds number problems here we present a novel coupling between an iterative force correction immersed boundary zhang et al 2016 and a multi domain cascaded lb method the iterative force correction immersed boundary method has been selected due to the improved accuracy of the computation while the cascaded lb formulation is used due to its superior stability at high reynolds numbers the coupling is shown to improve both the stability and numerical accuracy of the solution the resulting solver has been applied to viscous flow up to a reynolds number of 100000 passed a naca 0012 airfoil at a 10 degree angle of attack good agreement with results obtained using a body fitted navier stokes solver has been obtained the formulation provides a straight forward and efficient method for modeling realistic geometries and could easily be extended to problems with moving boundaries

on the benefits of laplace samples in solving a rare event problem using cross entropy method the convergence quality of the cross entropy ce optimizer relies critically on the mechanism meant for randomly generating data samples in agreement with the inference drawn in the earlier works the fast simulated annealing fsa and fast evolutionary programming fep since tracing a near global optimum embedded on a nonconvex search space can be viewed as a rare event problem a ce algorithm constructed using a long tailed distribution is intuitively attractive for effectively exploring the optimization landscape based on this supposition a set of ce algorithms employing the cauchy logistic and laplace distributions are experimentally validated in a wide range of optimization functions which are shifted rotated expanded and or composed characterized by convex unimodal discontinuous noisy and multimodal fitness landscapes the laplace distribution has been demonstrated to be more suitable for the ce optimization since the samples drawn have jump lengths long enough to elude local optima and short enough to preserve sufficient candidates in the global optimum neighborhood besides a theoretical analysis has been carried out to understand the following i benefits offered by the long tailed distributions towards evasion of local optima ii link between the variation in scale parameter estimate and the probability of producing candidate solutions arbitrarily close to the global optimum

the computation of strain rate tensor in multiple relaxation time lattice boltzmann model abstract the multiple relaxation time mrt lattice boltzmann lb model is an important class of lb model with lots of advantages over the traditional single relaxation time srt lb model generally the computation of strain rate tensor is crucial for the mrt lb simulations of some complex flows at present only two formulae are available to compute the strain rate tensor in the mrt lb model one is to compute the strain rate tensor using the non equilibrium parts of macroscopic moments yu formula the other is to compute the strain rate tensor using the non equilibrium parts of density distribution functions chai formula the mathematical expressions of these two formulae are so different that we do not know which formula to choose for computing the strain rate tensor in the mrt lb model to overcome this problem this paper presents a theoretical study of the relationship between chai and yu formulae the results show that the yu formula can be deduced from the chai formula although they have their own advantages and disadvantages in particular the yu formula is computationally more efficient while the chai formula is applicable to more lattice patterns of the mrt lb models furthermore the derivation of the yu formula in a particular lattice pattern from the chai formula is more convenient than that proposed by yu et al

modeled boltzmann equation and its application to direct aeroacoustics simulation the bhatnagar gross and krook modeled boltzmann equation has been applied to simulate different fluid dynamics problems with varying degrees of success however its application to direct aeroacoustic computation is less successful one possible reason could be its inability to recover the state equation correctly for a diatomic gas and hence an inaccurate determination of the speed of sound the present study reports on the development of an improved modeled boltzmann equation for aeroacoustics simulation the approach is to modify the maxwellian distribution normally assumed for the equilibrium particle distribution function constraints imposed are the exact recovery of the state equation for a diatomic gas and the euler equations without invoking the small mach number assumption thus formulated a distribution function consisting of the maxwellian distribution plus three other terms that attempt to account for particle particle collisions is obtained a velocity lattice method is used to solve the improved modeled boltzmann equation using an equivalent lattice equilibrium distribution function the simulations are validated against benchmark aeroacoustic problems whose solutions are deduced from a direct numerical simulation of the euler equations the results of the improved modeled boltzmann equation obtained using a smaller computational domain are in excellent agreement with those deduced from direct numerical simulation using a larger computational domain thus verifying the viability and correctness of the modified equilibrium distribution function

a characteristic galerkin method for discrete boltzmann equation abstract the characteristic galerkin finite element method for the discrete boltzmann equation is presented to simulate fluid flows in complex geometries the inherent geometric flexibility of the finite element method permits the easy use of simple cartesian variables on unstructured meshes and the mesh clustering near large gradients the characteristic galerkin procedure with appropriate boundary condition results in accurate solutions with little numerical diffusion several test cases are conducted including unsteady couette flows lid driven cavity flows and steady flow past a circular cylinder on unstructured meshes the numerical results are in good agreement with previous analytical if applicable numerical and experimental results

convergence of lattice boltzmann methods for navier stokes flows in periodic and bounded domains combining an asymptotic analysis of the lattice boltzmann method with a stability estimate we are able to prove some convergence results which establish a strict relation to the incompressible navier stokes equation the proof applies to the lattice boltzmann method in the case of periodic domains and for specific bounded domains if the dirichlet boundary condition is realized with the bounce back rule

unconventional initialization methods for differential evolution the crucial role played by the initial population in a population based heuristic optimization cannot be neglected it not only affects the search for several iterations but often also has an influence on the final solution if the initial population itself has some knowledge about the potential regions of the search domain then it is quite likely to accelerate the rate of convergence of the optimization algorithm in the present study we propose two schemes for generating the initial population of differential evolution de algorithm these schemes are based on quadratic interpolation qi and nonlinear simplex method nsm in conjugation with computer generated random numbers the idea is to construct a population that is biased towards the optimum solution right from the very beginning of the algorithm the corresponding algorithms named as qide using quadratic interpolation and nsde using non linear simplex method are tested on a set of 20 traditional benchmark problems with box constraints and 7 shifted non traditional functions taken from literature comparison of numerical results with traditional de and opposition based de ode show that the proposed schemes considered by us for generating the random numbers significantly improves the performance of de in terms of convergence rate and average cpu time

lattice boltzmann method for 3 d flows with curved boundary in this work we investigate two issues that are important to computational efficiency and reliability in fluid dynamic applications of the lattice boltzmann equation lbe 1 computational stability and accuracy of different lattice boltzmann models and 2 the treatment of the boundary conditions on curved solid boundaries and their 3 d implementations three athermal 3 d lbe models q15d3 q19d3 and q27d3 are studied and compared in terms of efficiency accuracy and robustness the boundary treatment recently developed by filippova and hanel 1998 j comp phys 147 219 and mei et al 1999 j comp phys 155 307 in 2 d is extended to and implemented for 3 d the convergence stability and computational efficiency of the 3 d lbe models with the boundary treatment for curved boundaries were tested in simulations of four 3 d flows 1 fully developed flows in a square duct 2 flow in a 3 d lid driven cavity 3 fully developed flows in a circular pipe and 4 a uniform flow over a sphere we found that while the 15 velocity 3 d q15d3 model is more prone to numerical instability and the q27d3 is more computationally intensive the q19d3 model provides a balance between computational reliability and efficiency through numerical simulations we demonstrated that the boundary treatment for 3 d arbitrary curved geometry has second order accuracy and possesses satisfactory stability characteristics

pyclaw accessible extensible scalable tools for wave propagation problems development of scientific software involves tradeoffs between ease of use generality and performance we describe the design of a general hyperbolic pde solver that can be operated with the convenience of matlab yet achieves efficiency near that of hand coded fortran and scales to the largest supercomputers this is achieved by using python for most of the code while employing automatically wrapped fortran kernels for computationally intensive routines and using python bindings to interface with a parallel computing library and other numerical packages the software described here is pyclaw a python based structured grid solver for general systems of hyperbolic pdes k t mandli et al pyclaw software version 1 0 http numerics kaust edu sa pyclaw 2011 pyclaw provides a powerful and intuitive interface to the algorithms of the existing fortran codes clawpack and sharpclaw simplifying code development and use while providing massive parallelism and scalable solvers via the petsc library the

derivation and analysis of lattice boltzmann schemes for the linearized euler equations we derive lattice boltzmann lbm schemes to solve the linearized euler equations in 1d 2d and 3d with the future goal of coupling them to an lbm scheme for navier stokes equations and a finite volume scheme for linearized euler equations the derivation uses the analytical maxwellian in a bgk model in this way we are able to obtain second order schemes in addition we perform an l 2 stability analysis numerical results validate the approach

equivalent partial differential equations of a lattice boltzmann scheme we show that when we formulate the lattice boltzmann equation with a small time step dt and an associated space scale dx a taylor expansion joined with the so called equivalent equation methodology leads to establishing macroscopic fluid equations as a formal limit we recover the euler equations of gas dynamics at the first order and the compressible navier stokes equations at the second order

a structured approach to the construction of stable linear lattice boltzmann collision operator abstract we introduce a structured approach to the construction of linear bgk type collision operators ensuring that the resulting lattice boltzmann methods are stable with respect to a weighted l 2 norm the results hold for particular boundary conditions including periodic bounce back and bounce back with flipping of sign boundary conditions this construction uses the equivalent moment space definition of bgk type collision operators and the notion of stability structures as guiding principle for the choice of the equilibrium moments for those moments influencing the error term only but not the order of consistency the presented structured approach is then applied to the 3d isothermal linearized euler equations with non vanishing background velocity finally convergence results in the strong discrete l norm highlight the suitability of the structured approach introduced in this manuscript

a smooth transition model between kinetic and hydrodynamic equations this paper presents a model which provides a smooth transition between a kinetic and a hydrodynamic domain the idea is to use a buffer zone in which both hydrodynamics and kinetic equations will be solved the solution of the original kinetic equation will be recovered as the sum of the solutions of these two equations we use an artificial connecting function which makes the equation on each domain degenerate at the end of the buffer zone thus no boundary condition is needed at the transition point consequently this model avoids the delicate issue of finding the interface condition in a typical domain decomposition method that couples a kinetic equation with hydrodynamic equations a simple kinetic scheme is developed to discretize our model and numerical examples are used to validate the method

modeled boltzmann equation and its application to shock capturing simulation a modified equilibrium distribution function for the bhatnagar gross krook type modeled boltzmann equation has recently been proposed the function was deduced using acoustics scaling to normalize the equation and allowed a correct recovery of similarly normalized euler equations it is a combination of a maxwellian distribution plus three other terms that are moments of particle velocity the lattice counterpart of the modified equilibrium distribution function also led to an exact recovery of the euler equations therefore there is no mach number limitation in the entire approach this lattice counterpart was able to replicate aeroacoustics problems involving vorticity acoustic and entropy acoustic interactions correctly and the simulations were carried out using a finite difference lattice boltzmann method employing only a two dimensional nine velocity lattice thus formulated the numerical scheme has no arbitrary constants and all calculations were carried out using one single relaxation time and a set of constants derived from the analysis this paper investigates the validity and extent of the formulation to capture shocks and resolve contact discontinuity and expansion waves in one and two dimensional riemann problems the simulations are carried out using the same two dimensional nine velocity lattice and identical set of constants and relaxation time they are compared with theoretical results and those obtained by solving the euler equations directly using harten s first order numerical scheme good agreement is obtained for all test cases however the modified equilibrium distribution function is not suitable for shock structure simulation for that an exact recovery of the navier stokes equations is required

carleson measures multipliers and integration operators for spaces of dirichlet type for 0 1 we let d p denote the space of those functions f which are analytic in the unit disc d z c z 1 and satisfy d 1 z 2 f z p dx dy in this paper we characterize the positive borel measures in d such that d p l q d 0 p q we also characterize the pointwise multipliers from d p to d q 0 p q i fp 2 p in particular we prove that if

cellular automaton fluids 1 basic theory continuum equations are derived for the large scale behavior of a class of cellular automaton models for fluids the cellular automata are discrete analogues of molecular dynamics in which particles with discrete velocities populate the links of a fixed array of sites kinetic equations for microscopic particle distributions are constructed hydrodynamic equations are then derived using the chapman enskog expansion slightly modified navier stokes equations are obtained in two and three dimensions with certain lattices viscosities and other transport coefficients are calculated using the boltzmann transport equation approximation some corrections to the equations of motion for cellular automaton fluids beyond the navier stokes order are given

asymptotic analysis of finite difference methods with this article we want to advocate the use of asymptotic methods for the analysis of finite difference schemes we present several examples to demonstrate the applicability of the approach advantages over the modified equation and truncation error analysis are pointed out