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 two dimensional unsteady burgers equation in this paper a special lattice boltzmann model is proposed to simulate two dimensional unsteady burgers equation the maximum principle and the stability are proved the model has been verified by several test examples excellent agreement is obtained between numerical predictions and exact solutions the cases of steep oblique shock waves are solved and compared with the two point compact scheme results the study indicates that lattice boltzmann model is highly stable and efficient even for the problems with severe gradients

thermodynamic foundations of kinetic theory and lattice boltzmann models for multiphase flows this paper demonstrates that thermodynamically consistent lattice boltzmann models for single component multiphase flows can be derived from a kinetic equation using both enskog s theory for dense fluids and mean field theory for long range molecular interaction the lattice boltzmann models derived this way satisfy the correct mass momentum and energy conservation equations all the thermodynamic variables in these lbm models are consistent the strengths and weaknesses of previous lattice boltzmann multiphase models are analyzed

a modified multiple relaxation time lattice boltzmann model for convection diffusion equation a modified lattice boltzmann model with multiple relaxation times mrt for the convection diffusion equation cde is proposed by modifying the relaxation matrix as well as choosing the corresponding equilibrium distribution function properly the present model can recover the cde with anisotropic diffusion coefficient with no deviation term even when the velocity vector varies generally with space or time through the chapman enskog analysis this model is firstly validated by simulating the diffusion of a gaussian hill which demonstrates it can handle the anisotropic diffusion problem correctly then it is adopted to calculate the longitudinal dispersion coefficient of the taylor aris dispersion numerical results show that the present model can further reduce the numerical error under the condition of non zero velocity vector especially when the dimensionless relaxation time is relatively large

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

introducing a new kinetic model which admits an h theorem for simulating the nearly incompressible fluid flows we present a new kinetic model for simulating the nearly incompressible fluid flows the model uses constant speed particles which are free to move in all directions in the new model the maxwellian is replaced by a fractional form while the navier stokes and continuity equations are recovered it is known that the conventional lattice boltzmann model lbm with polynomial equilibrium distribution function cannot admit an h theorem wagner 1998 9 in the present work we show that the new model admits an h theorem and the numerical schemes which stem from the new model are more stable than the conventional lbm for the streaming stage two different approaches namely the lattice based and the discontinuous galerkin based schemes are introduced the former is more cost effective than the conventional lattice boltzmann models while it maintains their simplicity the latter is a higher order spectral element method using which one can employ non uniform triangular elements with high accuracy and geometrical flexibility unlike the conventional lbm the new model is derived from continuous relations hence it does not require a symmetric discrete velocity model the accuracy and stability of the model have been verified by simulating three benchmark problems the plane couette flow lid driven square cavity and flow around an impulsively started cylinder have been simulated the results of the present work are in excellent agreement with the exact solutions and with the results reported by others

generalized petrov galerkin methods for the numerical solution of burgers equation generalized petrov galerkin methods are applied in two different ways for the numerical solution of burgers equation in the first place it is used to produce upwinding in the way described by mitchell and griffiths 9 10 the main objective of this paper is in the second place to use it as a moving finite element method in particular it is shown that the methods of miller and co workers4 6 and herbst et al 7 8 can be incorporated in the more general approach presented in this paper several choices for the test functions are discussed and numerical results are given

a lattice boltzmann equation for diffusion the formulation of lattice gas automata lga for given partial differential equations is not straightforward and still requires some sort of magic lattice boltzmann equation lbe models are much more flexible than lga because of the freedom in choosing equilibrium distributions with free parameters which can be set after a multiscale expansion according to certain requirements here a lbe is presented for diffusion in an arbitrary number of dimensions the model is probably the simplest lbe which can be formulated it is shown that the resulting algorithm with relaxation parameter 1 is identical to an explicit finite difference efd formulation at its stability limit underrelaxation 0 1 allows stable integration beyond the stability limit of efd the time step of the explicit lbe integration is limited by accuracy and not by stability requirements

a lattice boltzmann equation for waves we propose a lattice boltzmann model for the wave equation using a lattice boltzmann equation and the chapman enskog expansion we get 1d and 2d wave equations with truncation error of order two the numerical tests show the method can be used to simulate the wave motions

lattice boltzmann model for the modified burgers equation abstract in this paper a special lattice boltzmann model is developed to solve the modified burgers equation the maximum principle and the stability are proved the model has been verified by several test examples excellent agreement is obtained between numerical predictions and the constructed asymptotic solutions or available numerical results the careful numerical study indicates the computational efficiency and stability of the lattice boltzmann method

a comparison of differential quadrature methods for the solution of partial differential equations abstract discrete ordinate methods for the solution of partial differential equations are examined and compared a novel approach for the discrete representation of differential operators based on split range polynomial expansions is introduced the utility of the method is demonstrated for the case of differentiation of functions involving steep gradients the solution of burgers equation is presented to illustrate the effectiveness of the technique for the solution of nonlinear partial differential equations exhibiting nearly discontinuous solutions comparisons are made with other differential quadrature methods

a lattice boltzmann scheme for incompressible multiphase flow and its application in simulation of rayleigh taylor instability in this paper we propose a new lattice boltzmann scheme for simulation of multiphase flow in the nearly incompressible limit the new scheme simulates fluid flows based on distribution functions the interfacial dynamics such as phase segregation and surface tension are modeled by incorporating molecular interactions the lattice boltzmann equations are derived from the continuous boltzmann equation with appropriate approximations suitable for incompressible flow the numerical stability is improved by reducing the effect of numerical errors in calculation of molecular interactions an index function is used to track interfaces between different phases simulations of the two dimensional rayleigh taylor instability yield satisfactory results the interface thickness is maintained at 3 4 grid spacings throughout simulations without artificial reconstruction steps

a new approach to noise in quantum mechanics the standard way of describing noise in a quantum system consists in attaching to the system a reservoir or bath which is assumed to be in thermal equilibrium subsequently the combined equation of motion is solved to second order in the interaction and by averaging the result over the bath one gets the density matrix of the system itself however the differential equation obtained in this way has a serious flaw which can be attributed to the inappropriate initial condition for this reason we here take as starting point the thermal equilibrium of the combined system the averages and correlation functions of quantities of interest provide us with the required information about the noise this is explicitly demonstrated on the special model of a harmonic oscillator coupled to a bath of harmonic oscillators at temperature t the result is compared with the standard calculation and it is shown that the latter is incorrect for time intervals smaller than kt as an example the energy fluctuations in equilibrium are computed

a moving petrov galerkin method for transport equations a time dependent transport equation is solved numerically by using a petrov galerkin technique on a time dependent grid a piecewise linear trial solution and piecewise hermite cubic test functions are used the same method is also applied to a time independent problem in which case it automatically generates a grid in a way which bears some resemblance to certain existing grid generation techniques several numerical examples are given

moving mesh methods based on moving mesh partial differential equations several versions of a moving mesh method are developed based on a mesh spatial smoothing technique and on the moving mesh pdes derived in a previous paper these versions are quite simple and easy to program they are applied to three bench mark one dimensional problems which show different solution behaviour the numerical results clearly demonstrate that the present methods are capable of accurately tracking rapid spatial and temporal transitions

variational iteration method for solving burger s and coupled burger s equations by means of variational iteration method the solutions of burger s equation and coupled burger s equations are exactly obtained comparison with the adomian decomposition method is made showing that the former is more effective than the later in this paper he s variational iteration method is introduced to overcome the difficulty arising in calculating adomian polynomials

quadratic b spline finite element method for numerical solution of the burgers equation abstract in this paper the one dimensional nonlinear burgers equation was converted to a set of nonlinear ordinary differential equations by the method of discretization in time and then each of them was solved by applying the quadratic b spline finite element method in order to show the efficiency of the presented method two problems were considered the numerical solutions obtained for various values of viscosity were compared with the exact solutions it is seen that the results obtained by this way are quite satisfactory

a fluctuating lattice boltzmann scheme for the one dimensional kpz equation abstract based on the well known mapping between the burgers equation with noise and the kardar parisi zhang kpz equation for fluctuating interfaces we develop a fluctuating lattice boltzmann lb scheme for growth phenomena as described by the kpz formalism a very simple lb kpz scheme is demonstrated in 1 1 spacetime dimensions and is shown to reproduce the scaling exponents characterizing the growth of one dimensional fluctuating interfaces

a numerical study of three moving grid methods for one dimensional partial differential equations which are based on the method of lines in recent years several sophisticated packages based on the method of lines mol have been developed for the automatic numerical integration of time dependent problems in partial differential equations pdes notably for problems in one space dimension these packages greatly benefit from the very successful developments of automatic stiff ordinary differential equation solvers however from the pde point of view they integrate only in a semiautomatic way in the sense that they automatically adjust the time step sizes but use just a fixed space grid chosen a priori for the entire calculation for solutions possessing sharp spatial transitions that move e g travelling wave fronts or emerging boundary and interior layers a grid held fixed for the entire calculation is computationally inefficient since for a good solution this grid often must contain a very large number of nodes in such cases methods which attempt automatically to adjust the sizes of both the space and the time steps are likely to be more successful in efficiently resolving critical regions of high spatial and temporal activity methods and codes that operate this way belong to the realm of adaptive or moving grid methods following the mol approach this paper is devoted to an evaluation and comparison mainly based on extensive numerical tests of three moving grid methods for 1d problems viz the finite element method of miller and co workers the method published by petzold and a method based on ideas adopted from dorii and drury our examination of these three methods is aimed at assessing which is the most suitable from the point of view of retaining the acknowledged features of reliability robustness and efficiency of the conventional mol approach therefore considerable attention is paid to the temporal performance of the methods

numerical solutions of two dimensional burgers equations by discrete adomian decomposition method in this paper the discrete adomian decomposition method adm is proposed to numerically solve the two dimensional burgers nonlinear difference equations two test problems are considered to illustrate the accuracy of the proposed discrete decomposition method it is shown that the numerical results are in good agreement with the exact solutions for each problem

equidistributing principles in moving finite element methods abstract recently miller and his co workers proposed a moving finite element method based on a least squares principle this was followed by a similar method by the present authors using a petrov galerkin approach in this paper the two methods are compared in particular it is shown that both methods move their nodes according to an approximate equidistributing principle this observation leads to a criterion for the placement of the nodes it is also shown that the penalty function designed by miller may also be used with the petrov galerkin method finally numerical examples are given illustrating the performance of the two methods

homogeneous generalized master equations it is shown that the method proposed in v f los j phys a math gen 34 638 6403 2001 which allows for turning the inhomogeneous time convolution generalized master equation tc gme into homogeneous while retaining initial correlations time convolution generalized master equation tc hgme for the relevant part of a distribution function is fully applicable to the quantum case and to the time convolutionless gme tcl gme it is demonstrated by rederiving the tc hgme and showing that it works in both the classical and quantum physics cases the time convolutionless hgme tcl hgme retaining initial correlations which is formally the same for both the classical and quantum physics has also been derived both the tc hgme and tcl hgme are exact equations applicable on any timescale and allow for consecutive treating the initial correlations and collisions on the equal footing a new equation for a momentum distribution function retaining initial correlations has been obtained in the linear in the density of quantum particles approximation connection of this equation to the quantum boltzmann equation is discussed

a fully implicit finite difference scheme for two dimensional burgers equations the two dimensional burgers equations are discretized in fully implicit finite difference form this scheme leads to a system of nonlinear difference equations to be solved at each time step newton s method is used to solve this nonlinear system the linear system is solved by a direct method at each iteration of newton s method the accuracy of the proposed numerical scheme is examined by comparison with other analytical and numerical results the present method performs well

a new scheme for source term in lbgk model for convection diffusion equation a new scheme for source term in the lattice bgk lbgk model for convection diffusion equation is proposed unlike the models proposed previously the present scheme only requires the source term in order of the knudsen number by adding a differential operator of the source term to the evolution equation the scheme can be applied to reaction diffusion systems directly numerical results are found to be in excellent agreement with the analytical solutions it is also found that the numerical accuracy of the present scheme is generally much better than that of the existing models

global constraints for round robin tournament scheduling abstract in the presence of side constraints and optimization criteria round robin tournament problems are hard combinatorial problems commonly tackled with tree search and branch and bound optimization recent results indicate that constraint based tree search has crucial advantages over integer programming based tree search for this problem domain by exploiting global constraint propagation algorithms during search in this paper we analyze arc consistent propagation algorithms for the global constraints all different and one factor in the domain of round robin tournaments the best propagation algorithms allow us to compute all feasible perfectly mirrored pattern sets with minimal breaks for intermural tournaments of realistic size and to improve known lower bounds for intramural tournaments balanced with respect to carry over effects

a stable discretization of the lattice boltzmann equation for simulation of incompressible two phase flows at high density ratio a stable discretization of the lattice boltzmann equation lbe for non ideal gases is presented for simulation of incompressible two phase flows having high density and viscosity ratios the stiffness of the discretized forcing terms in lbe for non ideal gases is known to trigger severe numerical instability and restrict practical application of the lbe method use of a proper pressure updating scheme is also crucial to the stability of the lbe method because of non negligible pressure variation across the phase interface to deal with these issues we propose a stable discretization scheme which assumes the low mach number approximation and utilizes the stress and potential forms of the surface tension force the incompressible transformation and the consistent discretization of the intermolecular forcing terms the proposed stable discretization scheme is applied to simulate 1 d advection equation with a source term a stationary droplet droplet oscillation and droplet splashing and deposition on a thin film at a density ratio of 1000 with varying reynolds numbers the numerical solutions of stationary and oscillatory droplets agree well with analytic solutions including the laplace s law the time history of the spread factor of the liquid sheet emitted after the droplet impact also follows the known spreading power law

moving finite element model for one dimensional infiltration in unsaturated soil the numerical integration of richards equation for infiltration into homogeneous unsaturated soil is carried out by a moving finite element mfe method accurate solution of this one dimensional nonlinear partial differential equation by standard finite difference fd or finite element fe methods can be obtained by using a fine discretization on space and time this can lead to large computer times with the mfe method grid points are moved during computation along the wetting front so that accuracy can be maintained by using a small number of nodes for simulating cases several meters in depth for these cases the mfe method is faster than fd and fe methods the numerical mfe model is verified through comparison with the quasi analytical solution by philip 1957 moreover we compare for some other test problems the computational efficiency of this method with that of two efficient pressure head based fd and fe fixed grid formulations

convergence of lattice boltzmann methods for stokes flows in periodic and bounded domains combining an asymptotic analysis of the lattice boltzmann method m junk z yang asymptotic analysis of lattice boltzmann boundary conditions j stat phys 121 2005 3 35 with the stability estimate presented in m junk w a yong weighted l 2 stability of the lattice boltzmann equation preprint we are able to prove some strict convergence results the proof applies to the lattice boltzmann method with linear collision operator both in the case of periodic domains and bounded domains if the dirichlet boundary condition is realized with the bounce back rule

a lattice boltzmann model for elliptic equations with variable coefficient abstract a lattice boltzmann model is proposed to solve elliptic equations with variable coefficient compared with the previous model the present model is a more effective solver to the ellipse equation with variable coefficient and the ellipse equation is exactly recovered to order o e 2 the relaxation time is not fixed and is determined by the coefficient of the equation the limited numerical results show that the present model is valid for ellipse equations

logic based benders decomposition benders decomposition uses a strategy of learning from one s mistakes the aim of this paper is to extend this strategy to a much larger class of problems the key is to generalize the linear programming dual used in the classical method to an inference dual solution of the inference dual takes the form of a logical deduction that yields benders cuts the dual is therefore very different from other generalized duals that have been proposed the approach is illustrated by working out the details for propositional satisfiability and 0 1 programming problems computational tests are carried out for the latter but the most promising contribution of logic based benders may be to provide a framework for combining optimization and constraint programming methods

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

entropy function approach to the lattice boltzmann method in this paper we present the construction of the lattice boltzmann method equipped with the h theorem based on entropy functions whose local equilibria are suitable to recover the navier stokes equations in the framework of the lattice boltzmann method we derive a collision integral which enables simple identification of transport coefficients and which circumvents construction of the equilibrium we discuss performance of this approach as compared to the standard realizations

homogeneous generalized master equation retaining initial correlations using the projection operator technique the exact homogeneous generalized master equation hgme for the relevant part of a distribution function statistical operator is derived the exact mass operator governing the evolution of the relevant part of a distribution function and comprising arbitrary initial correlations is found neither the bogolyubov principle of weakening of initial correlations with time nor any other approximation such as random phase approximation has been used to obtain the hgme these approximations are usually used to derive the approximate homogeneous equation for the relevant part of a distribution function from the conventional exact generalized master equation gme which has a source containing the irrelevant part initial correlations the hgme does not have a source and contains only the linear relative to the relevant part of a distribution function terms of the gme modified by the dynamics of initial correlations the obtained equation is valid on any timescale for any initial moment of time and any initial correlations in particular it describes the short time behaviour and allows for treating the influence of initial correlations consistently as an example we have considered a dilute gas of classical particles by selecting the appropriate projection operator we have derived the homogeneous equation for a one particle distribution function retaining initial correlations in the linear approximation on the small density parameter and for the space homogeneous case this equation allows for considering all stages of the time evolution it converts into the conventional boltzmann equation on the appropriate timescale if the contribution of all initial correlations vanishes on this timescale

theory of the lattice boltzmann method three dimensional model for linear viscoelastic fluids a three dimensional lattice boltzmann model with thirty two discrete velocity distribution functions for viscoelastic fluid is presented in this work the model is based upon the generalized lattice boltzmann equation constructed in moment space the nonlinear equilibria of the model have a number of coupling constants that are free parameters the dispersion equation of the model is analyzed under various conditions to obtain the constraints on the free parameters such that the model satisfies isotropy and galilean invariance the macroscopic equations are also derived from the lattice boltzmann model through the dispersion equation analysis and the chapman enskog analysis we demonstrate that the dispersion equation analysis can be used as a general and effective means to derive hydrodynamic equations excluding some nonlinear source terms from the lattice boltzmann model to obtain conditions for its isotropy and galilean invariance and to optimize its stability we show that the hydrodynamic behavior of the lattice boltzmann model has memory effects and that in the linear regime it behaves as a viscoelastic fluid described by the jeffreys model some numerical results to verify the theoretical analysis of the model are also presented

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

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

scheduling double round robin tournaments with divisional play using constraint programming abstract we study a tournament format that extends a traditional double round robin format with divisional single round robin tournaments elitserien the top swedish handball league uses such a format for its league schedule we present a constraint programming model that characterizes the general double round robin plus divisional single round robin format this integrated model allows scheduling to be performed in a single step as opposed to common multistep approaches that decompose scheduling into smaller problems and possibly miss optimal solutions in addition to general constraints we introduce elitserien specific requirements for its tournament these general and league specific constraints allow us to identify implicit and symmetry breaking properties that reduce the time to solution from hours to seconds a scalability study of the number of teams shows that our approach is reasonably fast for even larger league sizes the experimental evaluation of the integrated approach takes considerably less computational effort to schedule elitserien than does the previous decomposed approach

numerical stability analysis of fdlbm we analyze the numerical stability of finite difference lattice boltzmann method fdlbm by means of von neumann stability analysis the stability boundary of the fdlbm depends on the bgk relaxation time the cfl number the mean flow velocity and the wavenumber as the bgk relaxation time is increased at constant cfl number the stability of the central difference lb scheme may not be ensured the limits of maximum stable velocity are obtained around 0 39 0 43 and 0 43 for the central difference for the explicit upwind difference and for the semi implicit upwind difference schemes respectively we derive artificial viscosities for every difference scheme and investigate their influence on numerical stability the requirements for artificial viscosity is consistent with the conditions derived from von neumann stability analysis this analysis elucidates that the upwind difference schemes are suitable for simulation of high reynolds number flows

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

derivation of the lattice boltzmann method by means of the discrete ordinate method for the boltzmann equation in this note we show that the lattice boltzmann bgk method in common use can be derived from the boltzmann equation itself according to the present derivation it is apparent that the equation thus derived can be reduced to the navier stokes equation at a limit of small knudsen number since the equation thus derived is the boltzmann equation itself hence the present derivation method gives us a simple and flexible recipe to construct the lattice boltzmann method and makes it easy to construct an extended lattice boltzmann method in various ways 15 refs 1 fig

a lattice boltzmann model for the generalized burgers huxley equation abstract in this paper we develop a lattice boltzmann model for the generalized burgers huxley equation gbhe by choosing the proper time and space scales and applying the chapman enskog expansion the governing equation is recovered correctly from the lattice boltzmann equation and the local equilibrium distribution functions are obtained excellent agreement with the exact solution is observed and better numerical accuracy is obtained than the available numerical result the results indicate the present model is satisfactory and efficient the method can also be applied to the generalized burgers fisher equation and be extended to multidimensional cases

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

moving finite element methods for evolutionary problems ii applications abstract in this the second of two papers on the subject we present applications of the moving finite element method to a number of test problems key features are linear elements a direct approach to parallelism and node overtaking avoiding penalty functions rapid inversion of the mass matrix by preconditioned conjugate gradients and explicit euler time stepping the resulting codes are fast and efficient and are able to follow fronts and similar features with great accuracy the paper includes a substantial section on changes of dependent variable and front tracking techniques for non linear diffusion problems test problems include non linear hyperbolic conservation laws and non linear parabolic equations in one and two dimensions

a benders approach for the constrained minimum break problem this paper presents a hybrid ip cp algorithm for designing a double round robin schedule with a minimal number of breaks both mirrored and non mirrored schedules with and without place constraints are considered the algorithm uses benders cuts to obtain feasible home away pattern sets in few iterations and this approach leads to significant reductions in computation time for hard instances furthermore the algorithm is capable of solving a number of previously unsolved benchmark problems for the traveling tournament problem with constant distances

lattice boltzmann model for generalized nonlinear wave equations in this paper a lattice boltzmann model is developed to solve a class of the nonlinear wave equations through selecting equilibrium distribution function and an amending function properly the governing evolution equation can be recovered correctly according to our proposed scheme in which the chapman enskog expansion is employed we validate the algorithm on some problems where analytic solutions are available including the second order telegraph equation the nonlinear klein gordon equation and the damped driven sine gordon equation it is found that the numerical results agree well with the analytic solutions which indicates that the present algorithm is very effective and can be used to solve more general nonlinear problems

lattice boltzmann models for two dimensional coupled burgers equations abstract in this paper two lattice boltzmann models for two dimensional coupled burgers equations are proposed through treating the part or all of convection items as the source term where the spatial gradient can be calculated by the distribution function the models can exactly recover the burgers equations without any assumptions some numerical tests are also performed to validate the present models it is found that the proposed models are more accurate and efficient in solving two dimensional coupled burgers equations

a new lattice boltzmann model for solving the coupled viscous burgers equation abstract in this paper a new lattice boltzmann model for the coupled nonlinear system of viscous burgers equation is proposed by using the double evolutionary equations through selecting equilibrium distribution functions and amending functions properly the governing evolution system can be recovered correctly according to our proposed scheme in which the chapman enskog expansion is employed the effects of space and time resolutions on the accuracy and stability of the model are numerically investigated in detail the numerical solutions for various initial and boundary conditions are calculated and validated against analytic solutions or other numerical solutions reported in previous studies it is found that the numerical results agree well with the analytic solutions which indicates the potential of the present algorithm for solving the coupled nonlinear system of viscous burgers equation

unified approach to simulation on deforming elements with application to phase change problems abstract several different approaches to simulation on deforming finite elements are shown to generate the same weighted residuals formulation for the evolution of the dependent variables control of node motion by means of mesh stretching in two dimensions is illustrated in the context of phase change problems simulation of problems with large phase boundary excursions shows good agreement with analytic solutions

mass transport theory for the toda lattices dispersive and dissipative abstract to establish mass transport theory on nonlinear lattices we formulate the korteweg devries kdv equation and the burgers equation using the flow variable representation so as to facilitate comparison with the boltzmann equation and with the cahn hilliard equation in classical statistical mechanics we also study toda lattice microdynamics using the flaschka representation and compare with the liouville equation like the linear diffusion equation the boltzmann equation and the liouville equation are to be solved for a distribution function which is intrinsically probabilistic transport theory in linear systems is governed by the isotropic motions of the kinetic equations in contrast the kdv perturbation equation derived from the toda lattice microdynamics expresses hydrodynamic mass transport the kdv equation in hydrodynamics and the burgers equation in thermodynamics do not involve a probability distribution function the nonlinear lattices do not retain isotropy of the mass transport equations in consequence it is proposed that in the presence of hydrodynamic flows to the left kdv wave propagation proceeds to the right this basic property of the kdv system is extended to thermodynamics in the burgers system these features arise because linear systems are driven towards an equilibrium by molecular collisions whereas the inhomogeneities of the nonlinear lattices are generated by the potential energy of interaction diffusion as expressed by the burgers equation is governed not only by a chemical potential but also by the toda lattice potential energy

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

a lattice boltzmann model for the generalized boussinesq equation a lattice boltzmann model is developed for the simulation of the generalized nonlinear boussinesq equation through adding a differential operator of the diffusion term as a source term to the evolution equation the macroscopic equation is recovered with higher order truncation error detailed numerical simulations for the motion of the soliton solutions of the boussinesq equation are performed and the numerical results agree well with the exact solutions the results show that the lattice boltzmann method is an efficient algorithm with excellent long time numerical behaviors for the motion of the soliton solutions

three dimensional multi relaxation time mrt lattice boltzmann models for multiphase flow in this paper three dimensional 3d multi relaxation time mrt lattice boltzmann lb models for multiphase flow are presented in contrast to the bhatnagar gross krook bgk model a widely employed kinetic model in mrt models the rates of relaxation processes owing to collisions of particle populations may be independently adjusted as a result the mrt models offer a significant improvement in numerical stability of the lb method for simulating fluids with lower viscosities we show through the chapman enskog multiscale analysis that the continuum limit behavior of 3d mrt lb models corresponds to that of the macroscopic dynamical equations for multiphase flow we extend the 3d mrt lb models developed to represent multiphase flow with reduced compressibility effects the multiphase models are evaluated by verifying the laplace young relation for static drops and the frequency of oscillations of drops the results show satisfactory agreement with available data and significant gains in numerical stability

microscopic origins of irreversible macroscopic behavior time asymmetric behavior as embodied in the second law of thermodynamics is observed in individual macroscopic systems it can be understood as arising naturally from time symmetric microscopic laws in accord with the ideas of thompson maxwell and boltzmann alternative explanations based on equating irreversible macroscopic behavior with the mixing type of behavior already present in the time evolution of ensembles probability distributions of certain systems having only a few degrees of freedom are in my opinion unnecessary misguided and misleading

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

an implementation of a moving finite element method abstract the moving finite element method an adaptive gridding procedure for systems of partial differential equations whose solutions contain steep gradients has been implemented in a very straightforward way accurate results can be obtained using fewer nodes than that required by fixed mesh solution methods performance of the method using some minor changes to a standard ordinary differential equation algorithm is illustrated with solutions to burger s equation

a branching scheme for finding cost minimal round robin tournaments abstract single round robin tournaments are a well known class of sports leagues schedules we consider leagues with a set t of n teams where n is even costs are associated to each possible match the goal is to find the minimum cost tournament among those having the minimum number of breaks we pick up structural properties of home away pattern sets having the minimum number of breaks a branching idea using these properties is developed in order to guide branching steps on the first levels of a branch and bound tree in order to avoid nodes corresponding to infeasible subproblems

a space time fully decoupled wavelet galerkin method for solving two dimensional burgers equations abstract a space time fully decoupled formulation for solving two dimensional burgers equations is proposed based on the coiflet type wavelet sampling approximation for a function defined on a bounded interval by applying a wavelet galerkin approach for spatial discretization nonlinear partial differential equations are first transformed into a system of ordinary differential equations in which all matrices are completely independent of time and never need to be updated in the time integration finally the mixed explicit implicit scheme is employed to solve the resulting semi discretization system by numerically studying three widely considered test problems results demonstrate that the proposed method has a much better accuracy and a faster convergence rate than many existing numerical methods most importantly the study also indicates that the present wavelet method is capable of solving the two dimensional burgers equation at high reynolds numbers

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

a new lattice boltzmann model for the laplace equation in this paper a new lattice boltzmann equation which is independent of time is proposed based on the new lattice boltzmann equation some steady problems can be modeled by the lattice boltzmann method in the further study the laplace equation is investigated with the method of the higher order moment of equilibrium distribution functions and a series of partial differential equations in different space scales the numerical results show that the new method is effective

the moving finite element method applications to general partial differential equations with multiple large gradients abstract the moving finite element mfe method has been reduced to practice in the automatic solution program dyla for general systems of transient partial differential equations pdes in 1 d several test examples are presented which illustrate the unique node movement and systematic control features which are intrinsic in the mfe method computational dilemmas of numerical diffusion gibbs overshooting and undershooting zone tangling and grid remap or re connection aliasing which occur frequently in conventional pde methods are essentially eliminated in the mfe method arbitrarily large gradients or shocks can be solved with extremely high resolution and accuracy for non coincident or even counterdirected propagating wavefronts boundary layers of arbitrarily small dimensions are solved with high accuracy simultaneously with the large scale structures in reactive and non reactive fluid calculations the mfe method requires a small fraction of the grid nodes which are used in conventional pde solution methods because the nodes migrate continuously and systematically to those positions where they are most needed in order to yield accurate pde solutions on entire problem domains courant friedrichs lewy time step limits are exceeded by wide margins by factors of two to several thousand finally the extension of the mfe method to 2 d is briefly discussed

lattice bhatnagar gross krook model for the lorenz attractor abstract in this paper we propose a new lattice bhatnagar gross krook model for simulation of the lorenz attractor the new model is based on a nonlinear diffusion reaction system we formulate a lattice bhatnagar gross krook model for the nonlinear diffusion reaction system by using a method of higher moments of the lattice boltzmann equation as a special case where the diffusion effect disappears we get the lorenz equation in the model we obtain a series of lattice boltzmann equations at different time scales and the conservation law at time scale t0 the equilibrium distribution functions are simpler than those for the standard lattice boltzmann model the numerical examples show that the method can be used to simulate the lorenz equation

macroscopic model and truncation error of discrete boltzmann method a derivation procedure to secure the macroscopically equivalent equation and its truncation error for discrete boltzmann method is proffered in this paper essential presumptions of two time scales and a small parameter in the chapman enskog expansion are disposed of in the present formulation equilibrium particle distribution function instead of its original non equilibrium form is chosen as key variable in the derivation route taylor series expansion encompassing fundamental algebraic manipulations is adequate to realize the macroscopically differential counterpart a self contained and comprehensive practice for the linear one dimensional convection diffusion equation is illustrated in details numerical validations on the incurred truncation error in one and two dimensional cases with various distribution functions are conducted to verify present formulation as shown in the computational results excellent agreement between numerical result and theoretical prediction are found in the test problems straightforward extensions to more complicated systems including convection diffusion reaction multi relaxation times in collision operator as well as multi dimensional navier stokes equations are also exposed in the appendix to point out its expediency in solving complicated flow problems

lattice boltzmann method for one and two dimensional burgers equation abstract in this paper a new method higher order moment lattice boltzmann method for one and two dimensional burgers equation is proposed the lattice boltzmann models presented here are based on a series of lattice boltzmann equations in different time scales in order to achieve higher order accuracy we use seven and four moments of the equilibrium distribution function in one and two dimensional models respectively we find two kinds of strategy to seek equilibrium distribution functions for the two dimensional model with second order accuracy these two are equivalent when a scale factor k 2 3 lastly we provide a fine numerical result of a one dimensional burgers equation numerical examples show the method can be used to simulate one and two dimensional burgers equation

parallel performance and accuracy of lattice boltzmann and traditional finite difference methods for solving the unsteady two dimensional burger s equation abstract lattice boltzmann methods are gaining recognition in the field of computational fluid dynamics due to their computational efficiency in order to quantify the computational efficiency and accuracy of the lattice boltzmann method it is compared with efficient traditional finite difference methods such as the alternating direction implicit scheme the lattice boltzmann algorithm implemented in previous studies does not approach peak performance for simulations where the data involved in computation per time step is more than the cache size due to this data is obtained from the main memory and this access is much slower than access to cache memory using a cache optimized lattice boltzmann algorithm this paper takes into account the full computational strength of the lattice boltzmann method the com parison is performed on both a single processor and multiple processors

a lattice boltzmann model for the nonlinear schrodinger equation the lattice boltzmann model for the nonlinear schrodinger equation is proposed the new model is based on the technique of the higher order moment of equilibrium distribution functions and a series of lattice boltzmann equations in different time scales the euler equations are derived from the nonlinear schrodinger equation by removing non physical pressure we have simulated two irrotational flows these numerical results agree well with classical ones

the modified local crank nicolson method for one and two dimensional burgers equations the modified local crank nicolson method is applied to solve one and two dimensional burgers equations new difference schemes that are explicit unconditionally stable and easy to compute are obtained numerical solutions obtained by the present method are compared with exact solutions and it is seen that they are in excellent agreement

a novel lattice boltzmann model for the coupled viscous burgers equations abstract in this work a new lattice boltzmann model for coupled burgers equations is proposed through selecting proper distribution functions unlike the previous models the present model can exactly recover the coupled equations without any assumptions a detailed numerical study on several coupled burgers equations is performed to validate the present model and the results show that the present model not only has a second order convergence rate in space but also is more accurate than the previous model

a multi round generalization of the traveling tournament problem and its application to japanese baseball abstract in a double round robin tournament involving n teams every team plays 2 n 1 games with one home game and one away game against each of the other n 1 teams given a symmetric n by n matrix representing the distances between each pair of home cities the traveling tournament problem ttp seeks to construct an optimal schedule that minimizes the sum total of distances traveled by the n teams as they move from city to city subject to several natural constraints to ensure balance and fairness in the ttp the number of rounds is set at r 2 in this paper we generalize the ttp to multiple rounds r 2 k for any k 1 and present an algorithm that converts the problem to finding the shortest path in a directed graph enabling us to apply dijkstra s algorithm to generate the optimal multi round schedule we apply our shortest path algorithm to optimize the league schedules for nippon professional baseball npb in japan where two leagues of n 6 teams play 40 sets of three intra league games over r 8 rounds our optimal schedules for the pacific and central leagues achieve a 25 reduction in total traveling distance compared to the 2010 npb schedule implying the potential for considerable savings in terms of time money and greenhouse gas emissions

numerical simulation for solitary wave of klein gordon zakharov equation based on the lattice boltzmann model abstract this work reports a study of klein gordon zakharov equation by using the lattice boltzmann method the klein gordon zakharov equation describes the interaction of the langmuir wave and the ion acoustic wave in plasma numerical simulations of the constructed lattice boltzmann model for the klein gordon zakharov equation show the propagation and the interaction of the 1 1 and 2 1 dimensional solitary waves these simulation results are examined and compared by using different parameters and different schemes this work indicates that the lattice boltzmann method can be a very effective tool for the simulation of klein gordon zakharov equation