comparisons of lattice boltzmann and finite difference methods for a two dimensional viscous burgers equation lattice boltzmann methods have been proposed as a computational means for solving various partial differential equations we look at three lattice boltzmann methods and an analogous finite difference method for solving a two dimensional scalar viscous burgers equation with periodic boundary conditions first using monotonicity arguments we prove that in the ell 1 norm the lattice boltzmann methods converge first order temporally and second order spatially then we provide some computational results which substantiate the theoretical results finally we compare the lattice boltzmann and finite difference methods the basis of comparison includes numerical results with regard to accuracy order of convergence performance and timing measurements memory requirements and conservations laws

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

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

grid refinement for lattice bgk models lattice boltzmann models proposed at the end of the 1980s as the noise free version of lattice gas models are based on gas kinetic representation of fluid flow their recent modifications the lattice bgk models provide especially simple effective and stable algorithms for the solution of hydrodynamical problems a local second order grid refinement scheme for the lattice bgk model is proposed in this work the refinement scheme and a boundary fitting scheme for complicated geometries are applied to simulate a benchmark problem of flow past a cylinder in a channel with small and moderate reynolds numbers

hydrodynamic behavior of lattice boltzmann and lattice bhatnagar gross krook models we present a numerical analysis of the ranges of validity of classical and generalized hydrodynamics for lattice boltzmann and lattice bhatnagar gross krook bgk methods in two and three dimensions as a function of the collision parameters of these models our analysis is based on the wave number dependence of the evolution operator good ranges of validity are found as long as the bgk relaxation time is chosen smaller than or equal to unity the additional degree of freedom in a two parameter bgk model can give further improvements

discretizations for the incompressible navier stokes equations based on the lattice boltzmann method a discrete velocity model with spatial and velocity discretization based on a lattice boltzmann method is considered in the low mach number limit a uniform numerical scheme for this model is investigated in the limit the scheme reduces to a finite difference scheme for the incompressible navier stokes equation which is a projection method with a second order spatial discretization on a regular grid the discretization is analyzed and the method is compared to chorin s original spatial discretization numerical results supporting the analytical statements are presented

a steady state lattice boltzmann model for incompressible flows in this paper we propose a steady state lattice boltzmann equation which is independent of time for steady incompressible flows on the basis of the steady state lattice boltzmann equation we find a way of modeling some steady state problems using the lattice boltzmann model in further study we investigate steady state incompressible flows with the method of the higher order moment of equilibrium distribution functions and a series of partial differential equations for different space scales by assuming r constant we obtain the momentum equation for incompressible steady state flow the numerical results show that the new method is effective

hamiltonian dynamics of homopolymer chain models the hamiltonian dynamics of chains of nonlinearly coupled particles is numerically investigated in two and three dimensions simple off lattice homopolymer models are used to represent the interparticle potentials time averages of observables numerically computed along dynamical trajectories are found to reproduce results given by the statistical mechanics of homopolymer models the dynamical treatment however indicates a nontrivial transition between regimes of slow and fast phase space mixing such a transition is inaccessible to a statistical mechanical treatment and reflects a bimodality in the relaxation of time averages to corresponding ensemble averages it is also found that a change in the energy dependence of the largest lyapunov exponent indicates the ensuremath theta transition between filamentary and globular polymer configurations clearly detecting the transition even for a finite number of particles

theory and applications of an alternative lattice boltzmann grid refinement algorithm this contribution proposes an alternative lattice boltzmann grid refinement algorithm that overcomes the drawbacks that plague existing approaches we demonstrate that this algorithm is accurate and applicable for all values of the relaxation time we also show that this algorithm can significantly speed up the flow settlement process by using a hierarchy of grid levels the stationary regime can be approached up to a thousand times faster than with a single grid resolution

regularity of minimizers of semilinear elliptic problems up to dimension 4 we consider the class of semistable solutions to semilinear equations u f u in a bounded smooth domain of input amssym bbb r n with convex in some results this class includes all local minimizers minimal and extremal solutions in dimensions n 4 we establish an a priori l bound that holds for every positive semistable solution and every nonlinearity f this estimate leads to the boundedness of all extremal solutions when n 4 and is convex this result was previously known only in dimensions n 3 by a result of g nedev in dimensions 5 n 9 the boundedness of all extremal solutions remains an open question it is only known to hold in the radial case br by a result of a capella and the author 2010 wiley periodicals inc

delaunay type singular solutions for the fractional yamabe problem we construct delaunay type solutions for the fractional yamabe problem with an isolated singularity begin aligned delta gamma w c n gamma w frac n 2 gamma n 2 gamma w 0 quad text in quad mathbb r n backslash 0 end aligned we follow a variational approach in which the key is the computation of the fractional laplacian in polar coordinates

fully lagrangian and lattice boltzmann methods for solving systems of conservation equations abstract a class of fully lagrangian methods for solving systems of conservation equations is defined the key step in formulating these methods is the definition of a new set of field variables for which lagrangian discretization is trivial recently popular lattice boltzmann simulation schemes for solving such systems are shown to be a useful sub class of these fully lagrangian methods in which a the conservation laws are satisfied at each grid point b the lagrangian variables are expanded perturbatively and c discretization error is used to represent physics such schemes are typically derived using methods of kinetic theory our numerical analysis approach shows that the conventional physical derivation while certainly valid and fruitful is not essential that it often confuses physics and numerics and that it can be unnecessarily constraining for example we show that lattice boltzmann like methods can be non perturbative and can be made higher order implicit and or with non uniform grids furthermore our approach provides new perspective on the relationship between lattice boltzmann methods and finite difference techniques among other things we show that the lattice boltzmann schemes are only conditionally consistent and in some cases are identical to the well known dufort frankel method through this connection the lattice boltzmann method provides a rational basis for understanding dufort frankel and gives a pathway for its generalization at the same time that dufort frankel is no longer much used suggests that the lattice boltzmann approach might also share this fate

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

regularity of radial minimizers and extremal solutions of semilinear elliptic equations we consider a special class of radial solutions of semilinear equations u g u in the unit ball of rn it is the class of semi stable solutions which includes local minimizers minimal solutions and extremal solutions we establish sharp pointwise lq and wk q estimates for semi stable radial solutions our regularity results do not depend on the specific nonlinearity g among other results we prove that every semi stable radial weak solution u h01 is bounded if n 9 for every g and belongs to h3 w3 2 in all dimensions n for every g increasing and convex the optimal regularity results are strongly related to an explicit exponent which is larger than the critical sobolev exponent

effect of the forcing term in the multiple relaxation time lattice boltzmann equation on the shear stress or the strain rate tensor in this work the effect of the forcing term or external force in the multiple relaxation time lattice boltzmann equation mrtlbe on the shear stress or the strain rate tensor is studied theoretically and numerically through a chapman enskog analysis and numerical simulations we show that the shear stress or the strain rate tensor derived from the mrtlbe is second order accurate in space we then examine the influence of the forcing term on the shear stress or the strain rate tensor and demonstrate that the forcing term effect must be included when the shear stress or the strain rate tensor is computed with the nonequilibrium part of the distribution function

truncation error analysis of lattice boltzmann methods a truncation error analysis is performed for models based on the lattice boltzmann lb equation this analysis involves two steps the recursive application of the lb equation and a taylor series expansion unlike previous analytical studies of lb methods the present work does not assume an asymptotic relationship between the temporal and spatial discretization parameters or between the probability distribution function f and its equilibrium distribution feq effective finite difference stencils are derived for both the distribution function and the primitive variables i e density and velocity the governing partial differential equations are also recovered the associated truncation errors are derived and the results are validated by numerical simulation of analytic flows analysis of the truncation errors elucidates the roles of the kinetic theory relaxation parameter and the discretization parameters x and t the effects of initial and boundary conditions are also addressed and are shown to significantly affect the overall accuracy of the method

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

simulation of cavity flow by the lattice boltzmann method a detailed analysis is presented to demonstrate the capabilities of the lattice boltzmann method thorough comparisons with other numerical solutions for the two dimensional driven cavity flow show that the lattice boltzmann method gives accurate results over a wide range of reynolds numbers studies of errors and convergence rates are carried out compressibility effects are quantified for different maximum velocities and parameter ranges are found for stable simulations the paper s objective is to stimulate further work using this relatively new approach for applied engineering problems in transport phenomena utilizing parallel computers

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

anomaly of the lattice boltzmann methods in three dimensional cylindrical flows truncation error terms of the lattice boltzmann equation lbe are analysed for three dimensional flow simulations by the d3q15 d3q19 and d3q27 models by applying the analysis of holdych et al 2004 9 rotational invariance of the reynolds averaged error terms is examined in cylindrical coordinates it is found that error terms distribute cyclically in the circumferential direction and produce false forces in particular directions by the d3q15 and d3q19 models when flow becomes turbulent this tendency is reinforced by the reynolds stress resulting in the generation of obvious spurious currents by the d3q27 model it is confirmed that the lbe keeps rotational invariance in laminar flows whilst non axisymmetric error terms arise in turbulent flows however the magnitudes of the error terms are less than two percent of those by the d3q15 and d3q19 models and hence unphysical spurious currents are sufficiently weak

efficient and accurate simulations of deformable particles immersed in a fluid using a combined immersed boundary lattice boltzmann finite element method the deformation of an initially spherical capsule freely suspended in simple shear flow can be computed analytically in the limit of small deformations d barthes biesel j m rallison the time dependent deformation of a capsule freely suspended in a linear shear flow j fluid mech 113 1981 251 267 those analytic approximations are used to study the influence of the mesh tessellation method the spatial resolution and the discrete delta function of the immersed boundary method on the numerical results obtained by a coupled immersed boundary lattice boltzmann finite element method for the description of the capsule membrane a finite element method and the skalak constitutive model r skalak a tozeren r p zarda s chien strain energy function of red blood cell membranes biophys j 13 1973 245 264 have been employed our primary goal is the investigation of the presented model for small resolutions to provide a sound basis for efficient but accurate simulations of multiple deformable particles immersed in a fluid we come to the conclusion that details of the membrane mesh as tessellation method and resolution play only a minor role the hydrodynamic resolution i e the width of the discrete delta function can significantly influence the accuracy of the simulations the discretization of the delta function introduces an artificial length scale which effectively changes the radius and the deformability of the capsule we discuss possibilities of reducing the computing time of simulations of deformable objects immersed in a fluid while maintaining high accuracy

on the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term we study questions of existence uniqueness and asymptotic behaviour for the solutions of u x t of the problem begin gathered text u t delta u lambda e u text lambda text 0 t 0 text x text varepsilon b hfill p text u x 0 u 0 x text x text varepsilon b hfill text u x t 0 text on text partial b times 0 infty hfill end gathered where b is the unit ball x varepsilon r n x text leqq text 1 text and n geqq 3 our interest is focused on the parameter 0 2 n 2 for which p admits a singular stationary solution of the form s x 2log x

regularity for the fractional gelfand problem up to dimension 7 abstract we study the problem s u e u in a bounded domain r n where is a positive parameter more precisely we study the regularity of the extremal solution to this problem our main result yields the boundedness of the extremal solution in dimensions n 7 for all s 0 1 whenever is for every i 1 n convex in the x i direction and symmetric with respect to x i 0 the same holds if n 8 and s 0 28206 or if n 9 and s 0 63237 these results are new even in the unit ball b 1

boundedness of extremal solutions in dimension 4 abstract in this paper we establish the boundedness of the extremal solution u in dimension n 4 of the semilinear elliptic equation u f u in a general smooth bounded domain r n with dirichlet data u 0 where f is a c 1 positive nondecreasing and convex function in 0 such that f s s as s in addition we prove that for n 5 the extremal solution u w 2 n n 2 this gives u l n n 4 if n 5 and u h 0 1 if n 6

direct assessment of lattice boltzmann hydrodynamics and boundary conditions for recirculating flows a hydrodynamic boundary condition is developed for lattice boltzmann hydrodynamics using a square orthogonal grid a constraint based on energy considerations is developed to provide closure for the equations which govern the particle distribution at the boundaries this boundary condition is applied to the two dimensional steady flow of an incompressible fluid behind a grid known as kovasznay flow the results are compared to those using alternate boundary conditions using the known exact solution the hydrodynamic boundary condition produces quadratic spatial convergence while alternate techniques fail to maintain this second order accuracy

optimal relaxation collisions for lattice boltzmann methods the optimal relaxation time of about 0 8090 has been proposed to balance the efficiency stability and accuracy at a given lattice size of numerical simulations with lattice boltzmann methods the optimal lattice size for a desired reynolds number can be refined by reducing the mach number for incompressible flows the functioned polylogarithm polynomials are defined and used to express the lattice boltzmann equations at different time scales and to analyze the impact of relaxation times and lattice sizes on truncation errors smaller truncation errors can be achieved when relaxation times are greater than 0 5 and less than 1 0 the steady state lid driven cavity flow was chosen to validate the code of lattice boltzmann procedures the applications of the optimal relaxation parameters numerically balance the stability efficiency and accuracy through hartmann flow the optimal relaxation time can also be used to select the initial lattice size for the channel flow over a square cylinder with a given mach number

lattice boltzmann method on composite grids a composite block structured lattice boltzmann method is proposed for the simulation of two dimensional incompressible fluid flows the grid structure is composed of a coarse base grid and one or several fine grid s the former covers the entire physical domain the latter are placed at regions where local grid refinement is desirable the simulation is first carried out on the base grid level at a smaller relaxation time allowing a rapid propagation of boundary information throughout the entire domain thus large scale flow features can be resolved efficiently at a relatively low cost at a later time fine grid variables are initiated the dependent variables on both grid levels are then advanced in time simultaneously with the fine grid boundary conditions obtained from the base grid solution at the grid interface as a demonstration the lid driven cavity flow is selected for study the results show good agreement with benchmark numerical data and those calculated from the finite volume u 2 mathrm rans model the proposed method is able to produce accurate solutions on fine grids with a considerable saving in cpu time

nonlocal elliptic equations involving measures abstract in this article we study the existence of solution for the problem u f u in u 0 in r n where 0 is a parameter 0 1 and is a radon measure a weak solution is obtained by using schauder s fixed point theorem in the case where is dirac measure the symmetry of the solution is obtained by using the moving plane method