a generalized finite difference gfd ale scheme for incompressible flows around moving solid bodies on hybrid meshfree cartesian grids a scheme using the mesh free generalized finite differencing gfd on flows past moving bodies is proposed the aim is to devise a method to simulate flow past an immersed moving body that avoids the intensive remeshing of the computational domain and minimizes data interpolation associated with the established computational fluid methodologies as such procedures are time consuming and are a significant source of error in flow simulation in the present scheme the moving body is embedded and enveloped by a cloud of mesh free nodes which convects with the motion of the body against a background of cartesian nodes the generalized finite difference gfd method with weighted least squares wls approximation is used to discretize the two dimensional viscous incompressible navier stokes equations at the mesh free nodes while standard finite difference approximations are applied elsewhere the convecting motion of the mesh free nodes is treated by the arbitrary lagrangian eulerian ale formulation of the flow equations which are solved by a second order crank nicolson based projection method the proposed numerical scheme was tested on a number of problems including the decaying vortex flow external flows past moving bodies and body driven flows in enclosures

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

a generalized lattice boltzmann model for fluid flow system and its application in two phase flows abstract in this paper a generalized lattice boltzmann lb model with a source term in the continuity equation is proposed to solve both incompressible and nearly incompressible navier stokes n s equations this model can be used to deal with single phase and two phase flows problems with a source term in the continuity equation from this generalized model we can not only get some existing models but also derive new models moreover for the incompressible model derived a modified pressure scheme is introduced to calculate the pressure and then to ensure the accuracy of the model in this work we will focus on a two phase flow system and in the frame work of our generalized lb model a new phase field based lb model is developed for incompressible and quasi incompressible two phase flows a series of numerical simulations of some classic physical problems including a spinodal decomposition a static droplet a layered poiseuille flow and a bubble rising flow under buoyancy are performed to validate the developed model besides some comparisons with previous quasi incompressible and incompressible lb models are also carried out and the results show that the present model is accurate in the study of two phase flows finally we also conduct a comparison between quasi incompressible and incompressible lb models for two phase flow problems and find that in some cases the proposed quasi incompressible lb model performs better than incompressible lb models

force evaluation in the lattice boltzmann method involving curved geometry the present work investigates two approaches for force evaluation in the lattice boltzmann equation the momentum exchange method and the stress integration method on the surface of a body the boundary condition for the particle distribution functions on curved geometries is handled with second order accuracy based on our recent works mei et al j comput phys 155 307 1999 ibid 161 680 2000 the stress integration method is computationally laborious for two dimensional flows and in general difficult to implement for three dimensional flows while the momentum exchange method is reliable accurate and easy to implement for both two dimensional and three dimensional flows several test cases are selected to evaluate the present methods including i two dimensional pressure driven channel flow ii two dimensional uniform flow past a column of cylinders iii two dimensional flow past a cylinder asymmetrically placed in a channel with vortex shedding iv three dimensional pressure driven flow in a circular pipe and v three dimensional flow past a sphere the drag evaluated by using the momentum exchange method agrees well with the exact or other published results

a nonlinearly preconditioned inexact newton algorithm for steady state lattice boltzmann equations most existing methods for calculating the steady state solution of the lattice boltzmann equations are based on pseudo time stepping which often requires a large number of time steps especially for high reynolds number problems to calculate the steady state solution directly without the time integration in this paper we propose and study a nonlinearly preconditioned inexact newton algorithm with a domain decomposition based linear solver for parallelization more precisely the proposed algorithmic framework involves an implicit second order discretization a two level inexact newton method and a nonlinear elimination preconditioner to accelerate the convergence of newton iteration a nonstandard pollution removing coarse space is introduced for the two level method numerical experiments are presented to demonstrate the robustness and efficiency of the algorithm especially for problems at a high reynolds number a comparison is also included to show the superiority of the proposed approach over o

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

acceleration of steady state lattice boltzmann simulations on non uniform mesh using local time step method in this paper the local time step method is applied to the lattice boltzmann method lbm on non uniform grid generalized form of interpolation supplemented lbm gilbm for steady flow simulations the local time step method is known to be an effective technique to accelerate the solution to the converged steady state on non uniform grid first code validation for global time step method is performed by solving flow around a cylinder at low reynolds numbers our results show good agreement with previous studies two dimensional flow simulations around an airfoil were performed in order to validate the code using the local time step method and the results are compared with the global time step solutions the pressure distribution and the aerodynamic coefficients of global and local time step results are in good agreement they are also consistent with the previous studies at the same time cpu time required to obtain the steady state solutions using the local time step are reduced by 70 80 compared with that using global time step

implicit velocity correction based immersed boundary lattice boltzmann method and its applications a version of immersed boundary lattice boltzmann method ib lbm is proposed in this work it is based on the lattice boltzmann equation with external forcing term proposed by guo et al z guo c zheng b shi discrete lattice effects on the forcing term in the lattice boltzmann method phys rev e 65 2002 046308 which can well consider the effect of external force to the momentum and momentum flux as well as the discrete lattice effect in this model the velocity is contributed by two parts one is from the density distribution function and can be termed as intermediate velocity and the other is from the external force and can be considered as velocity correction in the conventional ib lbm the force density external force is explicitly computed in advance as a result we cannot manipulate the velocity correction to enforce the non slip boundary condition at the boundary point in the present work the velocity corrections force density at all boundary points are considered as unknowns which are computed in such a way that the non slip boundary condition at the boundary points is enforced the solution procedure of present ib lbm is exactly the same as the conventional ib lbm except that the non slip boundary condition can be satisfied in the present model while it is only approximately satisfied in the conventional model numerical experiments for the flows around a circular cylinder and an airfoil show that there is no any penetration of streamlines to the solid body in the present results this is not the case for the results obtained by the conventional ib lbm another advantage of the present method is its simple calculation of force on the boundary the force can be directly calculated from the relationship between the velocity correction and the force density

on the orders of primitive groups with restricted nonabelian composition factors we prove that given c0 there exists c such that the following holds if g is a primitive permutation group of degree n no composition factor of which is an alternating group of degree greater than c0 or a classical group of dimension greater than c0 then g nc in particular if the nonabelian composition factors of g have bounded order then g is polynomially bounded

bases for primitive permutation groups and a conjecture of babai abstract a base of a permutation group g is a sequence b of points from the permutation domain such that only the identity of g fixes b pointwise we show that primitive permutation groups with no alternating composition factors of degree greater than d and no classical composition factors of rank greater than d have a base of size bounded above by a function of d this confirms a conjecture of babai

lattice boltzmann method for the convection diffusion equation in curvilinear coordinate systems a lattice boltzmann method for simulating convection and diffusion using a curvilinear grid system is presented the proposed method does not require an interpolation or coarse graining procedure and thus maintains the algorithmic simplicity of the original lattice boltzmann scheme the lattice boltzmann scheme is based on uniformly distributed lattice points in a transformed coordinate system and the apparent anisotropy of diffusion that arises due to the coordinate transformation is properly handled using the multiple relaxation time collision operator an asymptotic analysis of the lattice boltzmann equation shows that the proposed method appropriately reproduces the transformed convection diffusion equation several specific problems are numerically analyzed in order to validate the proposed method including an axially symmetric two dimensional problem in which the diffusion flux at an oblate hemispheroid is simulated using a body fitted orthogonal curvilinear grid system

lattice boltzmann accuracy in pore scale flow simulation we investigate the possibility of using nominally second order accurate techniques for resolving flow about solid boundaries as a means of improving accuracy and reducing grid resolution requirements in pore scale simulations an lbgk method is used to calculate flow in several geometries of increasing complexity using a first order accurate and two nominally second order accurate methods for no slip boundaries the geometries include uniform flow past an isolated sphere quadratic flow past a sphere near a wall flow through a bcc array of spheres and through a randomly packed bed of spheres the packed bed flows are also used to compare hydrodynamic dispersion results the results confirm second order accurate behavior where navier stokes flows are clearly developed however 3d pore scale simulations involve a trade off between resolution of the flow and the number of pore spaces and there is a resolution threshold below which certain flow features such as recirculation are not resolved we conjecture that most simulations will tend to operate near this threshold because of the competing demands for resolution and statistical accuracy we consider local flow features and the velocity distribution in addition to hydraulic permeability and drag to provide a fuller understanding of accuracy near this threshold

flow simulation around cambered airfoil by using conformal mapping and intermediate domain in lattice boltzmann method in this paper the developed interpolation lattice boltzmann method is used for simulation of unsteady fluid flow it combines the desirable features of the lattice boltzmann and the joukowski transformation methods this approach has capability to simulate flow around curved boundary geometries such as airfoils in a body fitted grid system simulation of unsteady flow around a cambered airfoil in a non uniform grid for the first time is considered to show the capability of this method for modeling of fluid flow around complex geometries and complicated long term periodic flow phenomena the developed solver is also coupled with a fast adaptive grid generator in addition the new approach retains all the advantages of the standard lattice boltzmann method the strouhal number the pressure the drag and the lift coefficients obtained from the simulations agree well with classical computational fluid dynamics simulations numerical studies for various test cases illustrate the strength of this new approach

spherical orbits and abelian ideals let g be a connected reductive complex algebraic group with lie algebra lie g g let b be a borel subgroup of g with unipotent radical bu we denote the lie algebra of b and bu by b and bu respectively the group b acts on any ideal of b by means of the adjoint representation after a preliminary section we study a relationship between spherical nilpotent orbits and abelian ideals a of b using the structure theory for these orbits from 10 the principal result of this section is that for an abelian ideal a of b any nilpotent orbit meeting a is a spherical g variety see theorem 2 3 as a consequence of this we obtain a short conceptual proof of a finiteness theorem from 14 namely for a parabolic subgroup p of g and an abelian ideal a of p lie p in the nilpotent radical pu lie pu the group p operates on a with finitely many orbits the proof of this fact in 14 involved long and tedious case by case considerations we also prove a partial converse to the result just mentioned following 4 we say that an ideal of b is ad nilpotent whenever it consists of nilpotent elements in case g is simply laced we show that an ad nilpotent ideal c of b is abelian provided any nilpotent orbit meeting c is spherical see proposition 2 7 in section 3 we consider some properties of ad nilpotent ideals of b in theorem 3 2 we give a description of the normaliser of such ideals this applies in particular to abelian ideals of b a remarkable theorem of doi 10 1006 aima 2000 1959 available online at http www idealibrary com on

abelian ideals of borel subalgebras and affine weyl groups abstract this paper is devoted to a detailed study of certain remarkable posets which form a natural partition of all abelian ideals of a borel subalgebra our main result is a nice uniform formula for the dimension of maximal ideals in these posets we also obtain results on ad nilpotent ideals which complete the analysis started in j algebra 225 2000 130 258 2002 112

flow simulation around an airfoil by lattice boltzmann method on generalized coordinates flow simulations around an airfoil naca0012 using lattice boltzmann method lbm are performed in this paper originally lbm is an incompressible flow solver on orthogonal coordinates however in order to resolve the boundary layer of the airfoil accurately the algorithm is extended to generalized coordinates also suitable wall boundary condition is introduced for generalized coordinates first of all the present numerical method is validated at relatively small reynolds number re 500 the velocity profile of the boundary layer is compared in detail with the cfl3d results and consistent results are obtained second the present method is combined with the baldwin lomax turbulence model and results at relatively high reynolds number re 5x10 5 are shown the results are in good agreement with the experimental data with the same grid resolution the present method can solve high reynolds

finite difference lattice boltzmann model for nonlinear convection diffusion equations in this paper a finite difference lattice boltzmann lb model for nonlinear isotropic and anisotropic convection diffusion equations is proposed in this model the equilibrium distribution function is delicately designed in order to recover the convection diffusion equation exactly different from the standard lb model the temporal and spatial steps in this model are decoupled such that it is convenient to study convection diffusion problem with the non uniform grid in addition it also preserves the advantage of standard lb model that the complex valued convection diffusion equation can be solved directly the von neumann stability analysis is conducted to discuss the stability region which can be used to determine the free parameters appeared in the model to test the performance of the model a series of numerical simulations of some classical problems including the diffusion equation the nonlinear heat conduction equation the sine gordon equation the gaussian hill problem the burgersfisher equation and the nonlinear schrdinger equation have also been carried out the results show that the present model has a second order convergence rate in space and generally it is also more accurate than the standard lb model

comparison of the lattice boltzmann method and the artificial compressibility method for navier stokes equations this paper compares in detail the lattice boltzmann method and an isothermal navier stokes method it is found that these two methods are closely related to each other both methods satisfy similar macroscopic governing equations in their continuous forms but they differ from each other in their discrete forms besides the obvious differences in stencils for spatial discretization these two methods also differ from each other in temporal discretization numerical tests show that these differences have little impact on the simulation of velocity fields but do generate noticeable differences in the pressure fields both methods are capable of simulating transient flows and exhibit oscillatory behavior due to the propagation of pressure waves the lattice boltzmann method may be more accurate for capturing the pressure waves

a simplified finite volume lattice boltzmann method for simulations of fluid flows from laminar to turbulent regime part i numerical framework and its application to laminar flow simulation abstract in this paper an unstructured grid based finite volume lattice boltzmann method fvlbm that can be used for the simulation of incompressible laminar flows is presented and studied in detail this method is derived from a simple modification of the cell vertex unstructured grid based fvlbm proposed by stiebler et al 2006 compared with other complex flux reconstruction methods the present scheme has a low computational cost and can also achieve second order spatial accuracy furthermore depending on the use of the different temporal discretization schemes the temporal accuracy can be adjusted for both steady and unsteady flows besides some comparisons of the computational cost and accuracy with another fvlbm scheme are also presented meanwhile different boundary conditions are illustrated that are easy to implement on complex geometries to validate the present method four cases are carried out including a couette flow driven by one plate for an accuracy test flow in a square cavity flow around a single circular cylinder and a more complex flow around double circular cylinders numerical experiments show that the present scheme can simulate steady and unsteady flows at relatively high reynolds number with relatively few grid cells thus demonstrating the good capability of the present method

base sizes of primitive groups bounds with explicit constants abstract we show that the minimal base size b g of a finite primitive permutation group g of degree n is at most 2 log g log n 24 this bound is asymptotically best possible since there exists a sequence of primitive permutation groups g of degrees n such that b g 2 log g log n 2 and b g is unbounded as a corollary we show that a primitive permutation group of degree n that does not contain the alternating group alt n has a base of size at most max n 25

a stochastic boundary forcing for dissipative particle dynamics the method of dissipative particle dynamics dpd is an effective coarse grained model of the hydrodynamics of complex fluids dpd simulations of wall bounded flows are however often associated with spurious fluctuations of the fluid properties near the wall we present a novel stochastic boundary forcing for dpd simulations of wall bounded flows based on the identification of fluctuations in simulations of the corresponding homogeneous system at equilibrium the present method is shown to enforce accurately the no slip boundary condition while minimizing spurious fluctuations of material properties in a number of benchmark problems

acceleration of lattice bgk schemes with grid refinement abstract the lbgk method with local grid refinement has been shown to be an efficient and accurate tool for the simulation of incompressible viscous flows over complex geometries in the present study further improvement of this concept is proposed enabling the use of smaller amount of time steps on refined grids without impairing the spatial or temporal accuracy this extension of the lbgk method has been proved by analytical and numerical investigations the gain in computational time was found to be significant

on pyber s base size conjecture let g be a permutation group on a finite set a subset of is a base for g if its pointwise stabilizer in g is trivial the base size of g denoted b g is the smallest size of a base a well known conjecture of pyber from the early 1990s asserts that there exists an absolute constant c such that b g c log g logn for any primitive permutation group g of degree n several special cases have been verified in recent years including the almost simple and diagonal cases in this paper we prove pyber s conjecture for all non affine primitive groups

the base size of a primitive diagonal group abstract a base b for a finite permutation group g acting on a set is a subset of with the property that only the identity of g can fix every point of b we prove that a primitive diagonal group g has a base of size 2 unless the top group of g is the alternating or symmetric group acting naturally in which case the minimal base size of g is determined up to two possible values we also prove that the minimal base size of g satisfies a well known conjecture of pyber moreover we prove that if the top group of g does not contain the alternating group then the proportion of pairs of points that are bases for g tends to 1 as g tends to infinity a similar result for the case when the degree of the top group is fixed is given

lattice boltzmann method on curvilinear coordinates system using an interpolation based strategy the lattice boltzmann method is extended to apply to general curvilinear coordinate systems as an example a cylindrical coordinate system is used to simulate two dimensional flow around a circular cylinder numerical simulations are carried out for impulsive initial conditions with reynolds numbers up to 104 the agreement of our results with previous computational and experimental results is satisfactory compared with previous lattice boltzmann simulations of the same problem our new approach greatly enhances the computational efficiency

ad nilpotent ideals of a borel subalgebra ii we provide an explicit bijection between the ad nilpotent ideals of a borel subalgebra of a simple lie algebra g and the orbits of q h 1 q under the weyl group q being the coroot lattice and h the coxeter number of g from this result we deduce in a uniform way a counting formula for the ad nilpotent ideals

implicit heat flux correction based immersed boundary finite volume method for thermal flows with neumann boundary conditions abstract in the frameworks of immersed boundary method ibm and finite volume method fvm an implicit heat flux correction based ib fvm is proposed for thermal flows with neumann boundary conditions with the use of a fractional step technique the preconditioned navier stokes n s equations are solved by the fvm to obtain the intermediate solution in the prediction step and the heat flux is corrected by enforcing the neumann condition in the correction step different from existing ibms the cell face centers are defined as the eulerian points due to the heat flux computation at each face in the fvm the neumann condition is implemented in such a way that the interpolated temperature gradient is equal to the specified boundary value at the same point when the corrected gradient field is interpolated from the face centers to the lagrangian points to achieve an implicit algorithm the temperature derivative corrections at the lagrangian points are set as unknowns and a system of algebraic equations is established by constructing hybrid thin plate splines tps interpolation delta function distribution in the derivative interpolation process the much more accurate tps is introduced because the use of cosine delta function yields a less accurate solution after the distribution process the heat flux correction of a fluid cell is evaluated by using the solved temperature derivative corrections at the face centers but that of a solid cell is calculated by using their additive inverses to supplement the same amount of heat flux into the solid domain as that flowing into the fluid domain across the boundary finally the heat flux of a cell is corrected by adding the correction to the intermediate value and the corrected heat flux is utilized to solve the n s equations in the prediction step as compared with the available implicit ibms for neumann conditions the present method avoids the introduction of auxiliary layers of lagrangian points as well as the approximate conversion from the neumann to dirichlet condition and thus is suitable for an arbitrary geometry the proposed method is verified by simulating several benchmark thermal flows with neumann conditions the natural convection in an annulus and the steady or unsteady forced convection over a stationary or oscillating cylinder all the computed results agree well with the literature data

casimir elements associated with levi subalgebras of simple lie algebras and their applications abstract let g be a simple lie algebra h a levi subalgebra and c h u h the casimir element defined via the restriction of the killing form on g to h we study c h eigenvalues in g h and related h modules without loss of generality one may assume that h is a maximal levi then g is equipped with the natural z grading g i z g i such that g 0 h and g i is a simple h module for i 0 we give explicit formulae for the c h eigenvalues in each g i i 0 and relate eigenvalues of c h in g 1 to the dimensions of abelian subspaces of g 1 then we prove that if a g 1 is abelian whereas g 1 is not then dim a dim g 1 2 moreover if dim dim g 1 2 then a has an abelian complement the z gradings of height 2 are closely related to involutions of g and we provide a connection of our theory to an extension of the strange formula of freudenthal de vries

spectral element discontinuous galerkin lattice boltzmann simulation of flow past two cylinders in tandem with an exponential time integrator in this paper a spectral element discontinuous galerkin sedg lattice boltzmann discretization and an exponential time marching scheme are used to study the flow field past two circular cylinders in tandem arrangement the basic idea is to discretize the streaming step of the lattice boltzmann equation by using the sedg method to get a system of ordinary differential equations odes whose exact solutions are expressed by using a large matrix exponential the approximate solution of the resulting odes are obtained from a projection method based on a krylov subspace approximation this approach allows us to approximate the matrix exponential of a very large and sparse matrix by using a matrix of much smaller dimension the exponential time integration scheme is useful especially when computations are carried out at high courant friedrichs lewy cfl numbers where most explicit time marching schemes are inaccurate simulations of flow were carried out for a circular cylinder at re 20 and for two circular cylinders in tandem at re 40 and a spacing of 2 5d where d is the diameter of the cylinders we compare our results with those from a fourth order runge kutta scheme that is restricted by the cfl number in addition important flow parameters such as the drag coefficients of the two cylinders and the wake length behind the rear cylinder were calculated by using the exponential time integration scheme these results are compared with results from our simulation using the rk scheme and with existing benchmark results

bases of primitive linear groups abstract we correct and improve a result in 3 giving the structure of finite primitive linear groups of unbounded base size this confirms a well known conjecture of pyber on base sizes of primitive permutation groups in the case of affine groups whose associated linear group is primitive

a proof of pyber s base size conjecture abstract building on earlier papers of several authors we establish that there exists a universal constant c 0 such that the minimal base size b g of a primitive permutation group g of degree n satisfies log g log n b g 45 log g log n c this finishes the proof of pyber s base size conjecture the main part of our paper is to prove this statement for affine permutation groups g v h where h g l v is an imprimitive linear group an ingredient of the proof is that for the distinguishing number d g in the sense of albertson and collins of a transitive permutation group g of degree n 1 we have the estimates g n d g 48 g n

some progress in lattice boltzmann method part i nonuniform mesh grids a new lattice boltzmann algorithm is proposed to simulate the navier stokes equation on arbitrary nonuniform mesh grids the new algorithm retains the advantages of the lattice boltzmann method parallel of algorithm ease of programming and ability to incorporate microscopic interactions a simulation of flow in a two dimensional symmetric channel with sudden expansion is carried out using the new algorithm on a nonuniform mesh the results of the simulation are in excellent agreement with previous experimental and numerical results

a solution adaptive lattice boltzmann method for two dimensional incompressible viscous flows a stencil adaptive lattice boltzmann method lbm is developed in this paper it incorporates the stencil adaptive algorithm developed by ding and shu 26 for the solution of navier stokes n s equations into the lbm calculation based on the uniform mesh the stencil adaptive algorithm refines the mesh by two types of 5 points symmetric stencils which are used in an alternating sequence for increased refinement levels the two types of symmetric stencils can be easily combined to form a 9 points symmetric structure using the one dimensional second order interpolation recently developed by wu and shu 27 along the straight line and the d2q9 model the adaptive lbm calculation can be effectively carried out note that the interpolation coefficients are only related to the lattice velocity and stencil size hence the simplicity of lbm is not broken down and the accuracy is maintained due to the use of adaptive technique much less mesh points are required in the simulation as compared to the standard lbm as a consequence the computational efficiency is greatly enhanced the numerical simulation of two dimensional lid driven cavity flows is carried out accurate results and improved efficiency are reached in addition the steady and unsteady flows over a circular cylinder are simulated to demonstrate the capability of proposed method for handling problems with curved boundaries the obtained results compare well with data in the literature

a transformation free hoc scheme for incompressible viscous flows past an impulsively started circular cylinder in this paper we present a higher order compact scheme for the unsteady two dimensional 2d navier stokes equations on nonuniform polar grids specifically designed for the incompressible viscous flows past a circular cylinder the scheme is second order accurate in time and at least third order accurate in space the scheme very efficiently computes both unsteady and time marching steady state flow for a wide range of reynolds numbers re ranging from 10 to 9500 for the impulsively started cylinder the robustness of the scheme is highlighted when it accurately captures the vortex shedding for moderate re represented by the von karman street and the so called and s phenomena for higher re comparisons are made with established numerical and experimental results and excellent agreement is found in all the cases both qualitatively and quantitatively

an eulerian description of the streaming process in the lattice boltzmann equation this paper presents a novel strategy for solving discrete boltzmann equation dbe for simulation of fluid flows this strategy splits the solution procedure into streaming and collision steps as in the lattice boltzmann equation lbe method the streaming step can then be carried out by solving pure linear advection equations in an eulerian framework this offers two significant advantages over previous methods first the relationship between the relaxation parameter and the discretization of the collision term developed from the lbe method is directly applicable to the dbe method the resulting dbe collision step remains local and poses no constraint on time step second decoupling of the advection step from the collision step facilitates implicit discretization of the advection equation on arbitrary meshes an implicit unstructured dbe method is constructed based on this strategy and is evaluated using several test cases of flow over a backward facing step lid driven cavity flow and flow past a circular cylinder the speedup of convergence for some cases improves by a factor of about 20

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

a hybrid wavelet based adaptive immersed boundary finite difference lattice boltzmann method for two dimensional fluid structure interaction abstract a second generation wavelet based adaptive finite difference lattice boltzmann method fd lbm is developed in this paper in this approach the adaptive wavelet collocation method awcm is firstly to the best of our knowledge incorporated into the fd lbm according to the grid refinement criterion based on the wavelet amplitudes of density distribution functions an adaptive sparse grid is generated by the omission and addition of collocation points on the sparse grid the finite differences are used to approximate the derivatives to eliminate the special treatments in using the fd based derivative approximation near boundaries the immersed boundary method ibm is also introduced into fd lbm by using the adaptive technique the adaptive code requires much less grid points as compared to the uniform mesh code as a consequence the computational efficiency can be improved to justify the proposed method a series of test cases including fixed boundary cases and moving boundary cases are invested a good agreement between the present results and the data in previous literatures is obtained which demonstrates the accuracy and effectiveness of the present awcm ib lbm

abelian ideals of a borel subalgebra and subsets of the dynkin diagram let g be a simple lie algebra and ab g the set of abelian ideals of a borel subalgebra of g in this note an interesting connection between ab g and the subsets of the dynkin diagram of g is discussed we notice that the number of abelian ideals with k generators equals the number of subsets of the dynkin diagram with k connected components for g of type an or cn we provide a combinatorial explanation of this coincidence by constructing a suitable bijection we also construct a general bijection between ab g and the subsets of the dynkin diagram which is based on the theory developed by peterson and kostant

on commutative polarizations let l be a finite dimensional lie algebra over a field k of characteristic zero and let u l be its enveloping algebra with quotient division ring d l let p be a commutative lie subalgebra of l in o2 the necessary and sufficient condition on p was given in order for d p to be a maximal commutative subfield of d l in particular this condition is satisfied if p is a commutative polarization cp with respect to any regular f l and the converse holds if l is ad algebraic the purpose of this paper is to study lie algebras admitting these cp s and to demonstrate their widespread occurrence first we have the following characterisation if l is completely solvable p is a cp of l if and only if there exists a descending chain of lie subalgebras

the lattice boltzmann equation on irregular lattices a general framework to extend the lattice boltzmann equation to arbitrary lattice geometries is presented and numerically demonstrated for the case of a two dimensional poiseuille flow the new scheme considerably extends the range of applicability of the boltzmann method to problems requiring the use of nonuniform grids

distributed lagrange multiplier fictitious domain method in the framework of lattice boltzmann method for fluid structure interactions a new implementation of the lattice boltzmann method lbm for fluid structure interactions is presented the idea of the distributed lagrange multiplier fictitious domain method dlm fd is introduced in the framework of the lattice boltzmann algorithm this implementation employs a fixed mesh for the solution of the fluid problem and a lagrangian formulation for the solid problem the main advantage of the method is that the re meshing procedure normally required in the ale method is circumvented numerical examples are provided to verify the algorithm and illustrate the capacity of the method to deal with the fluid elastic solid interactions

lattice bgk model for incompressible navier stokes equation abstract most of the existing lattice boltzmann bgk models lbgk can be viewed as compressible schemes to simulate incompressible fluid flows the compressible effect might lead to some undesirable errors in numerical simulations in this paper a lbgk model without compressible effect is designed for simulating incompressible flows the incompressible navier stokes equations are exactly recovered from this incompressible lbgk model numerical simulations of the plane poiseuille flow the unsteady 2 d shear decaying flow the driven cavity flow and the flow around a circular cylinder are performed the results agree well with the analytic solutions and the results of previous studies

base sizes for simple groups and a conjecture of cameron let g be a permutation group on a finite set a base for g is a subset b with pointwise stabilizer in g that is trivial we write b g for the smallest size of a base for g in this paper we prove that b g 6 if g is an almost simple group of exceptional lie type and is a primitive faithful g set an important consequence of this result when combined with other recent work is that b g 7 for any almost simple group g in a non standard action proving a conjecture of cameron the proof is probabilistic and uses bounds on fixed point ratios

finite volume versus streaming based lattice boltzmann algorithm for fluid dynamics simulations a one to one accuracy and performance study a finite volume fv discretization method for the lattice boltzmann lb equation which combines high accuracy with limited computational cost is presented in order to assess the performance of the fv method we carry out a systematic comparison focused on accuracy and computational performances with the standard streaming lattice boltzmann equation algorithm in particular we aim at clarifying whether and in which conditions the proposed algorithm and more generally any fv algorithm can be taken as the method of choice in fluid dynamics lb simulations for this reason the comparative analysis is further extended to the case of realistic flows in particular thermally driven flows in turbulent conditions we report the successful simulation of high rayleigh number convective flow performed by a lattice boltzmann fv based algorithm with wall grid refinement

faces and maximizer subsets of highest weight modules abstract in this paper we study general highest weight modules v over a complex finite dimensional semisimple lie algebra g we present three formulas for the set of weights of a large family of modules v which include but are not restricted to all simple modules and all parabolic verma modules these formulas are direct and do not involve cancellations and were not previously known in the literature our results extend the notion of the weyl polytope to general highest weight g modules v we also show that for all simple modules the convex hull of the weights is a w j invariant polyhedron for some parabolic subgroup w j we compute its vertices faces and symmetries more generally we also do this for all parabolic verma modules and for all modules v with highest weight not on a simple root hyperplane to show our results we extend the notion of convexity to arbitrary additive subgroups a r of coefficients our techniques enable us to completely classify weak a faces of the support sets wt v in the process extending classical results of satake borel tits vinberg and casselman as well as modern variants by chari dolbin ridenour and cellini marietti to general highest weight modules

on the finite difference based lattice boltzmann method in curvilinear coordinates the lattice boltzmann method is a microscopic based approach for solving the fluid flow problems at the macroscopic scales the presently popular method uses regularly spaced lattices and cannot handle curved boundaries with desirable flexibility to circumvent such difficulties a finite difference based lattice boltzmann method fdlbm in curvilinear coordinates is explored using body fitted coordinates with non uniform grids several test cases including the impulsively started cylindrical couette flow steady state cylindrical couette flow steady flow over flat plates and steady flow over a circular cylinder are used to examine various issues related to the fdlbm the effect of boundary conditions for the distribution functions on the solution the merits between second order central difference and upwind schemes for advection terms and the effect of the reynolds number are investigated favorable results are obtained using fdlbm in curvilinear coordinates indicating that the method is potentially capable of solving finite reynolds number flow problems in complex geometries

simulation of flows around an impulsively started circular cylinder by taylor series expansion and least squares based lattice boltzmann method the two dimensional incompressible viscous flow past an impulsively started circular cylinder for a wide range of reynolds numbers re 20 9500 is studied computationally by using an explicit taylor series expansion and least squares based lattice boltzmann method the final equation for distribution function in our method is in an explicit form and essentially has no limitation on choice of mesh structure and lattice model it can be easily applied to simulation of flows with curved boundaries such as the problem considered in this work for the flow past an impulsively started circular cylinder numerical results obtained by present method agree very well with experimental data and computational results of navier stokes equations available in the literature

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

an investigation into curved and moving boundary treatments in the lattice boltzmann method curved boundary treatments provide a means of improving the computational accuracy of the conventional stair shaped approximation used in lattice boltzmann lb simulations furthermore curved boundary treatments can be extended to the modeling of moving boundary problems simply by adding a momentum term to the bounced distribution functions at the solid surface this study commences by investigating three conventional interpolating treatments for curved boundaries in lb problems namely the filippova and hanel fh model o filippova d hanel grid refinement for lattice bgk models j comput phys 147 1998 219 228 bouzidi s model m bouzidi m firdaouss p lallemand momentum transfer of a boltzmann lattice fluid with boundaries phys fluids 13 11 2001 3452 3459 and yu s model d yu r mei w shyy a unified boundary treatment in lattice boltzmann method aiaa 2003 0953 new york 2003 previous investigations have indicated that the interpolations would break the mass conservation at the boundaries since the inaccuracy in evaluation of the momentum transfer at boundary leads to a net mass flux based on this reason a concept of the interpolation free treatment for modeling the curved and moving boundary conditions is proposed to overcome the drawback of these interpolation based curved boundary treatments in present study two interpolation free models are then proposed namely on site interpolation free osif and composite interpolation free cpif models these proposed models are initially applied to simulate the flow in the channels containing a stationary square block positioned at various locations along the longitudinal axis the simulations results are then compared with those obtained using the three conventional interpolating treatments the interpolation free models are then applied to the case of moving boundary problems in which a square block and a cylindrical block respectively move with a constant speed along a channel containing stationary flows to test the galilean effect of the proposed cpif model a couette flow past the stationary square cylinder block with the moving top bottom walls is simulated overall the numerical results show that the proposed interpolation free curved treatment models significantly improve the accuracy of the mass flux computation near the solid surface and thus enhance the accuracy of the momentum interaction at the moving boundaries

lattice boltzmann equation on a two dimensional rectangular grid abstract we construct a multirelaxation lattice boltzmann model 1 on a two dimensional rectangular grid the model is partly inspired by a previous work of koelman 2 to construct a lattice bgk model on a two dimensional rectangular grid the linearized dispersion equation is analyzed to obtain the constraints on the isotropy of the transport coefficients and galilean invariance for various wave propagations in the model the linear stability of the model is also studied the model is numerically tested for three cases a a vortex moving with a constant velocity on a mesh with periodic boundary conditions b poiseuille flow with an arbitrary inclined angle with respect to the lattice orientation and c a cylinder asymmetrically placed in a channel the numerical results of these tests are compared with either analytic solutions or the results obtained by other methods satisfactory results are obtained for the numerical simulations

regular article an accurate curved boundary treatment in the lattice boltzmann method the lattice boltzmann equation lbe is an alternative kinetic method capable of solving hydrodynamics for various systems major advantages of the method are due to the fact that the solution for the particle distribution functions is explicit easy to implement and natural to parallelize because the method often uses uniform regular cartesian lattices in space curved boundaries are often approximated by a series of stairs that leads to reduction in computational accuracy in this work a second order accurate treatment of the boundary condition in the lbe method is developed for a curved boundary the proposed treatment of the curved boundaries is an improvement of a scheme due to o filippova and d hanel 1998 j comput phys 147 219 the proposed treatment for curved boundaries is tested against several flow problems 2 d channel flows with constant and oscillating pressure gradients for which analytic solutions are known flow due to an impulsively started wall lid driven square cavity flow and uniform flow over a column of circular cylinders the second order accuracy is observed with a solid boundary arbitrarily placed between lattice nodes the proposed boundary condition has well behaved stability characteristics when the relaxation time is close to 1 2 the zero limit of viscosity the improvement can make a substantial contribution toward simulating practical fluid flow problems using the lattice boltzmann method

investigation of stability and hydrodynamics of different lattice boltzmann models stability and hydrodynamic behaviors of different lattice boltzmann models including the lattice boltzmann equation lbe the differential lattice boltzmann equation dlbe the interpolation supplemented lattice boltzmann method islbm and the taylor series expansion and least square based lattice boltzmann method tllbm are studied in detail our work is based on the von neumann linearized stability analysis under a uniform flow condition the local stability and hydrodynamic dissipation behaviors are studied by solving the evolution operator of the linearized lattice boltzmann equations numerically our investigation shows that the lbe schemes with interpolations such as dlbe islbm and tllbm improve the numerical stability by increasing hyper viscosities at large wave numbers small scales it was found that these interpolated lbe schemes with the upwind interpolations are more stable than those with central interpolations because of much larger hyper viscosities

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

generators and commutators in finite groups abstract quotients of compact groups the first part of the paper establishes results about products of commutators in a d generator finite group g for example if h g g1 gr then every element of the subgroup h g is a product of f r factors of the form h 1 g 1 h 1 prime g 1 1 ldots lbrack h r g r h r prime g r 1 with h 1 h 1 prime ldots allowbreak h r h r prime in h under certain conditions on h a similar conclusion holds with the significantly weaker hypothesis that g h g1 gr where f r is replaced by f1 d r the results are applied in the second part of the paper to the study of normal subgroups in finitely generated profinite groups and in more general compact groups results include the characterization of topologically finitely generated compact groups which have a countably infinite image and of those which have a virtually dense normal subgroup of infinite index as a corollary it is deduced that a compact group cannot have a finitely generated infinite abstract quotient

application of multi block approach in the immersed boundary lattice boltzmann method for viscous fluid flows the immersed boundary lattice boltzmann method was presented recently to simulate the rigid particle motion it combines the desirable features of the lattice boltzmann and immersed boundary methods it uses a regular eulerian grid for the flow domain and a lagrangian grid for the boundary for the lattice boltzmann method as compared with the single relaxation time collision scheme the multi relaxation time collision scheme has better computational stability due to separation of the relaxations of various kinetic models especially near the geometric singularity so the multi relaxation time collision scheme is used to replace the single relaxation time collision scheme in the original immersed boundary lattice boltzmann method in order to obtain an accurate result very fine lattice grid is needed near the solid boundary to reduce the computational effort local grid refinement is adopted to offer high resolution near a solid body and to place the outer boundary far away from the body so the multi block scheme with the multi relaxation time collision model is used in the immersed boundary lattice boltzmann method in each block uniform lattice spacing can still be used in order to validate the multi block approach for the immersed boundary lattice boltzmann method with multi relaxation time collision scheme the numerical simulations of steady and unsteady flows past a circular cylinder and airfoil are carried out and good results are obtained

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

a transformation free ψ v formulation of the navier stokes equations on compact nonuniform grids abstract in this work we have developed a second order accurate compact finite difference scheme for the biharmonic form of the steady state navier stokes n s equations on nonuniform cartesian grids without transformation contrary to the schemes developed on rectangular cartesian grids for n s equations that could handle only rectangular boundaries the proposed scheme can easily accommodate bluff bodies with curved boundary the scheme is applied to one problem having analytical solution and three fluid flow problems having different complexities while one of them viz the lid driven cavity depict internal flows the flow past a circular cylinder and a flat plate immersed in uniform flow represents external flows the scheme is shown to handle both dirichlet and neumann boundary conditions with equal ease the efficiency of the scheme is demonstrated not only by its ability to capture smaller scales up to post quaternary level of vortices for the flow in lid driven cavity but also its robustness in handling non rectangular boundaries of immersed bodies while the scheme has handled the boundary of an immersed body not necessarily aligned to the grid lines for one of the problems for another it has tackled a curved boundary on cartesian grid without the need of roping in any immersed interface in the process its efficiency is further asserted by the proximity of the computed solutions to available experimental results

least squares finite element scheme for the lattice boltzmann method on an unstructured mesh a numerical model of the lattice boltzmann method lbm utilizing least squares finite element method in space and the crank nicolson method in time is developed this method is able to solve fluid flow in domains that contain complex or irregular geometric boundaries by using the flexibility and numerical stability of a finite element method while employing accurate least squares optimization fourth order accuracy in space and second order accuracy in time are derived for a pure advection equation on a uniform mesh while high stability is implied from a von neumann linearized stability analysis implemented on unstructured mesh through an innovative element by element approach the proposed method requires fewer grid points and less memory compared to traditional lbm accurate numerical results are presented through two dimensional incompressible poiseuille flow couette flow and flow past a circular cylinder finally the proposed method is applied to estimate the permeability of a randomly generated porous media which further demonstrates its inherent geometric flexibility

a novel immersed boundary velocity correction lattice boltzmann method and its application to simulate flow past a circular cylinder a novel immersed boundary velocity correction lattice boltzmann method is presented and validated in this work by its application to simulate the two dimensional flow over a circular cylinder the present approach is inspired from the conventional immersed boundary method ibm in the conventional ibm the effect of rigid body on the surrounding flow is modeled through a forcing term which is in turn used to correct the surrounding velocity field it was found that this process is actually an iterative procedure trying to satisfy the non slip boundary condition at the solid wall in this work a new concept of immersed boundary velocity correction approach is proposed which directly corrects the velocity to enforce the physical boundary condition the main advantage of the new method is that it is simple in concept and easy for implementation and the convergence of numerical computation is faster and more stable than the conventional ibm one challenging issue of conventional ibm is that some streamlines may pass through the solid body since there is no mechanism to enforce the non slip condition at the boundary as shown in the present numerical results this unphysical phenomenon is avoided in our new method since the non slip condition is enforced the present results for the steady and unsteady flows compare very well with available data in the literature

flow past an impulsively started circular cylinder using a higher order semicompact scheme in the present investigation an attempt has been made to solve the two dimensional incompressible viscous flow past an impulsively started circular cylinder for reynolds numbers ranging from 20 to 5000 using higher order semicompact scheme hosc unlike conventional higher order compact schemes the hosc scheme has been developed to handle the circular geometry of the chosen problem and the intensive algebraic manipulations have been reduced considerably by relaxing the compactness of the computational stencil for few terms but retained for most of the terms of the discretized equations for the flow past an impulsively started circular cylinder the results obtained at low and moderate reynolds numbers have been validated with the experimental and numerical observations available in the literature for high reynolds number flows the present scheme rightly captures the alpha phenomenon at re 1000 and both beta and alpha phenomena one after the other at re 5000

least squares finite element lattice boltzmann method a new numerical model of the lattice boltzmann method utilizing least squares finite element in space and crank nicolson method in time is presented the new method is able to solve problem domains that contain complex or irregular geometric boundaries by using finite element method s geometric flexibility and numerical stability while employing efficient and accurate least squares optimization for the pure advection equation on a uniform mesh the proposed method provides for fourth order accuracy in space and second order accuracy in time with unconditional stability in the time domain accurate numerical results are presented through two dimensional incompressible poiseuille flow and couette flow

long abelian ideals abstract we study abelian ideals of a borel subalgebra consisting of long roots it is shown that methods of cellini and papi can be extended to this situation a uniform expression for the number of long abelian ideals is given we also show that there is a one to one correspondence between the long abelian ideals and b stable commutative subalgebras in the little adjoint representation of the langlands dual lie algebra

a faster isomorphism test for graphs of small degree in a recent breakthrough babai proceedings of stoc acm new york 2016 pp 684 697 gave a quasipolynomial time graph isomorphism test in this work we give an improved isomorphism test for g

on finitely generated profinite groups i strong completeness and uniform bounds we prove that in every finitely generated profinite group every subgroup of finite index is open this implies that the topology on such groups is determined by the algebraic structure this is deduced from the main result about finite groups let w be a locally finite group word and d n then there exists f f w d such that in every d generator finite group g every element of the verbal subgroup w g is equal to a product of f w values an analogous theorem is proved for commutators this implies that in every finitely generated profinite group each term of the lower central series is closed the proofs rely on some properties of the finite simple groups to be established in part ii

lattice boltzmann simulations of 2d laminar flows past two tandem cylinders we apply the lattice boltzmann equation lbe with multiple relaxation time mrt collision model to simulate laminar flows in two dimensions 2d in order to simulate flows in an unbounded domain with the lbe method we need to address two issues stretched non uniform mesh and inflow and outflow boundary conditions we use the interpolated grid stretching method to address the need of non uniform mesh we demonstrate that various inflow and outflow boundary conditions can be easily and consistently realized with the mrt lbe the mrt lbe with non uniform stretched grids is first validated with a number of test cases the poiseuille flow the flow past a cylinder asymmetrically placed in a channel and the flow past a cylinder in an unbounded domain we use the lbe method to simulate the flow past two tandem cylinders in an unbounded domain with re 100 our results agree well with existing ones through this work we demonstrate the effectiveness of the mrt lbe method with grid stretching

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

abelian ideals in a borel subalgebra of a complex simple lie algebra let mathfrak g be a complex simple lie algebra and mathfrak b a fixed borel subalgebra of mathfrak g we describe the abelian ideals in mathfrak b in a uniform way that is independent of the classification of complex simple lie algebras as an application we derive a formula for the maximal dimension of a commutative lie subalgebra of mathfrak g

a discontinuous petrov galerkin methodology for adaptive solutions to the incompressible navier stokes equations the discontinuous petrov galerkin methodology with optimal test functions dpg of demkowicz and gopalakrishnan 18 20 guarantees the optimality of the solution in an energy norm and provides several features facilitating adaptive schemes whereas bubnov galerkin methods use identical trial and test spaces petrov galerkin methods allow these function spaces to differ in dpg test functions are computed on the fly and are chosen to realize the supremum in the inf sup condition the method is equivalent to a minimum residual method for well posed problems with sufficiently regular solutions dpg can be shown to converge at optimal rates the inf sup constants governing the convergence are mesh independent and of the same order as those governing the continuous problem 48 dpg also provides an accurate mechanism for measuring the error and this can be used to drive adaptive mesh refinements we employ dpg to solve the steady incompressible navier stokes equations in two dimensions building on previous work on the stokes equations and focusing particularly on the usefulness of the approach for automatic adaptivity starting from a coarse mesh we apply our approach to a manufactured solution due to kovasznay as well as the lid driven cavity flow backward facing step and flow past a cylinder problems

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