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

quantum ergodicity for graphs related to interval maps we prove quantum ergodicity for a family of graphs that are obtained from ergodic one dimensional maps of an interval using a procedure introduced by pakonski et al j phys a 34 9303 9317 2001 as observables we take the l 2 functions on the interval the proof is based on the periodic orbit expansion of a majorant of the quantum variance specifically given a one dimensional lebesgue measure preserving map of an interval we consider an increasingly refined sequence of partitions of the interval to this sequence we associate a sequence of graphs whose directed edges correspond to elements of the partitions and on which the classical dynamics approximates the perron frobenius operator corresponding to the map we show that except possibly for subsequences of density 0 the eigenstates of the quantum graphs equidistribute in the limit of large graphs

a semi analytical iterative technique for solving nonlinear problems we present a semi analytical iterative method for solving nonlinear differential equations to demonstrate the working of the method we consider some nonlinear ordinary differential equations with appropriate initial boundary conditions in each of the examples we demonstrate the accuracy and convergence of the method to the solution we demonstrate clearly that the method is accurate fast and has a high order of convergence

stabilization of the lattice boltzmann method using the ehrenfests coarse graining idea the lattice boltzmann method lbm and its variants have emerged as promising computationally efficient and increasingly popular numerical methods for modeling complex fluid flow however it is acknowledged that the method can demonstrate numerical instabilities e g in the vicinity of shocks we propose a simple technique to stabilize the lbm by monitoring the difference between microscopic and macroscopic entropy populations are returned to their equilibrium states if a threshold value is exceeded we coin the name ehrenfests steps for this procedure in homage to the vehicle that we use to introduce the procedure namely the ehrenfests coarse graining idea

optimal stability of advection diffusion lattice boltzmann models with two relaxation times for positive negative equilibrium despite the growing popularity of lattice boltzmann schemes for describing multi dimensional flow and transport governed by non linear anisotropic advection diffusion equations there are very few analytical results on their stability even for the isotropic linear equation in this paper the optimal two relaxation time otrt model is defined along with necessary and sufficient easy to use von neumann stability conditions for a very general anisotropic advection diffusion equilibrium in one to three dimensions with or without numerical diffusion quite remarkably the otrt stability bounds are the same for any peclet number and they are defined by the adjustable equilibrium parameters such optimal stability is reached owing to the free kinetic relaxation parameter furthermore the sufficient stability bounds tolerate negative equilibrium functions the distribution divided by the local mass often labeled as unphysical we prove that the non negativity condition is i a sufficient stability condition of the trt model with any eigenvalues for the pure diffusion equation ii a sufficient stability condition of its otrt and bgk srt sub classes for any linear anisotropic advection diffusion equation and iii unnecessarily more restrictive for any peclet number than the optimal sufficient conditions adequate choices of the two relaxation rates and the free tunable equilibrium parameters make the otrt sub class more efficient than the bgk one at least in the advection dominant regime and allow larger time steps than known criteria of the forward time central finite difference schemes ftcs mftcs for both advection and diffusion dominant regimes

diagonal approximation of the form factor of the unitary group the form factor of the unitary group u n endowed with the haar measure characterizes the correlations within the spectrum of a typical unitary matrix it can be decomposed into a sum over pairs of periodic orbits where by periodic orbit we understand any sequence of matrix indices from here the diagonal approximation can be defined in the usual fashion as a sum only over pairs of identical orbits we prove that as we take the dimension n to infinity the diagonal approximation becomes exact that is converges to the full form factor in the interval 0 1

universal spectral form factor for chaotic dynamics we consider the semiclassical limit of the spectral form factor k of fully chaotic dynamics starting from the gutzwiller type double sum over classical periodic orbits we set out to recover the universal behaviour predicted by random matrix theory both for dynamics with and without time reversal invariance for times smaller than half the heisenberg time th f 1 we extend the previously known expansion to include the cubic term beyond confirming the random matrix behaviour of individual spectra the virtue of that extension is that the diagrammatic rules come in sight which determine the families of orbit pairs responsible for all orders of the expansion

explicit spectral formulas for scaling quantum graphs we present an exact analytical solution of the spectral problem of quasi one dimensional scaling quantum graphs strongly stochastic in the classical limit these systems are frequently employed as models of quantum chaos we show that despite their classical stochasticity all scaling quantum graphs are explicitly solvable in the form en fsnd where n is the sequence number of the energy level of the quantum graph and f is a known function which depends only on the physical and geometrical properties of the quantum graph our method of solution motivates a new classification scheme for quantum graphs we show that each quantum graph can be uniquely assigned an integer m reflecting its level of complexity we show that a network of taut strings with piecewise constant mass density provides an experimentally realizable analogue system of scaling quantum graphs

lattice boltzmann method for compressible flows with high mach numbers in this paper we present a lattice boltzmann model to simulate compressible flows by introducing an attractive force this scheme has two main advantages one is to soften sound speed effectively which greatly raises the mach number up to 5 another is its relative simple procedure simulations of the march cone and the comparison between theoretical expectations and simulations demonstrate that the scheme is effective in the simulation of compressible flows with high mach numbers which would create many new applications

action correlation of orbits through non conventional time reversal recently sieber and richter calculated semiclassically a first off diagonal contribution to the orthogonal form factor for a billiard on a surface of constant negative curvature following prior suggestions from the theory of disordered systems they considered orbit pairs with almost the same action for a generalization to systems invariant under non conventional time reversal which also belong to the orthogonal symmetry class we show here that it is necessary to redefine the configuration space by a suitable canonical transformation the distinction of this space is that it lets time reversal look conventional

coupled parallel flow through composite porous layers in this work we consider fluid flow through composite porous layers the flow through the layers is either governed by the same flow model with different permeabilities or by different models in both cases a matching condition at the interface between the layers is employed in the numerical solution of the flow problem finite difference expressions are derived for the interfacial velocity and take into account the local truncation error which is expressed as a function of the flow parameters and the step size employed

on the form factor for the unitary group we study the combinatorics of the contributions to the form factor of the group u n in the large n limit this relates to questions about semiclassical contributions to the form factor of quantum systems described by the unitary ensemble

on the level spacing distribution in quantum graphs we derive a formula for the level spacing probability distribution in quantum graphs we apply it to simple examples and we discuss its relation with previous work and its possible application in more general cases moreover we derive an exact and explicit formula for the level spacing distribution of integrable quantum graphs

no quantum ergodicity for star graphs we investigate statistical properties of the eigenfunctions of the schrodinger operator on families of star graphs with incommensurate bond lengths we show that these eigenfunctions are not quantum ergodic in the limit as the number of bonds tends to infinity by finding an observable for which the quantum matrix elements do not converge to the classical average we further show that for a given fixed graph there are subsequences of eigenfunctions which localise on pairs of bonds we describe how to construct such subsequences explicitly these structures are analogous to scars on short unstable periodic orbits

unitary stochastic matrix ensembles and spectral statistics we propose to study unitary matrix ensembles defined in terms of unitary stochastic transition matrices associated with markov processes on graphs we argue that the spectral statistics of such an ensemble after ensemble averaging depends crucially on the spectral gap between the leading and subleading eigenvalue of the underlying transition matrix it is conjectured that unitary stochastic ensembles follow one of the three standard ensembles of random matrix theory in the limit of infinite matrix size n if the spectral gap of the corresponding transition matrices closes slower than 1 n the hypothesis is tested by considering several model systems ranging from binary graphs to uniformly and non uniformly connected star graphs and diffusive networks in arbitrary dimensions

combinatorial theory of the semiclassical evaluation of transport moments i equivalence with the random matrix approach to study electronic transport through chaotic quantum dots there are two main theoretical approachs one involves substituting the quantum system with a random scattering matrix and performing appropriate ensemble averaging the other treats the transport in the semiclassical approximation and studies correlations among sets of classical trajectories there are established evaluation procedures within the semiclassical evaluation that for several linear and non linear transport moments to which they were applied have always resulted in the agreement with random matrix predictions we prove that this agreement is universal any semiclassical evaluation within the accepted procedures is equivalent to the evaluation within random matrix theory the equivalence is shown by developing a combinatorial interpretation of the trajectory sets as ribbon graphs maps with certain properties and exhibiting systematic cancellations among their contributions remaining trajectory sets can be identified with primitive palindromic factorisations whose number gives the coefficients in the corresponding expansion of the moments of random matrices the equivalence is proved for systems with and without time reversal symmetry

truncation effect on taylor aris dispersion in lattice boltzmann schemes the taylor dispersion in parabolic velocity field provides a well known benchmark for advection diffusion ade schemes and serves as a first step towards accurate modeling of the high order non gaussian effects in heterogeneous flow while applying the lattice boltzmann ade two relaxation times trt scheme for a transport with given peclet number pe one should select six free tunable parameters namely i molecular diffusion scale equilibrium parameter ii three families of equilibrium weights assigned to the terms of mass velocity and numerical diffusion correction and iii two relaxation rates we analytically and numerically investigate the respective roles of all these degrees of freedom in the accuracy and stability in the evolution of a gaussian plume for this purpose the third and fourth order transient multi dimensional analysis of the recurrence equations of the trt ade scheme is extended for a spatially variable velocity field the key point is in the coupling of the truncation and taylor dispersion analysis which allows us to identify the second order numerical correction k t to taylor dispersivity coefficient k t the procedure is exemplified for a straight poiseuille flow where k t is given in a closed analytical form in equilibrium and relaxation parameter spaces the predicted longitudinal dispersivity is in excellent agreement with the numerical experiments over a wide parameter range in relatively small pe range the relative dispersion error increases with peclet number this deficiency reduces in the intermediate and high pe range where it becomes pe independent and velocity amplitude independent eliminating k t by a proper parameter choice and employing specular reflection for zero flux condition on solid boundaries the d2q9 trt ade scheme may reproduce the taylor aris result quasi exactly from very coarse to fine grids and from very small to arbitrarily high peclet numbers since free tunable product of two eigenfunctions also controls stability of the model the validity of the analytically established von neumann stability diagram is examined in poiseuille profile the simplest coordinate stencil subclass which is the d2q5 trt bounce back scheme demonstrates the best performance and achieves the maximum accuracy for most stable relaxation parameters closed analytical form is derived for truncation dispersivity of the d2q9 advection diffusion lbm scheme in higher peclet range its relative contribution to taylor result becomes pe and velocity independent numerical dispersivity vanishes for equilibrium velocity weight dependent eigenfunction product in trt collision analytically predicted longitudinal dispersivity is in excellent agreement with the numerical experiments the d2q5 trt bounce back scheme achieves maximum accuracy for most stable relaxation parameters

on the search of more stable second order lattice boltzmann schemes in confined flows the von neumann linear analysis restricted by a heuristic selection of wave number vectors was applied to the search of explicit lattice boltzmann schemes which exhibit more stability than existing methods the relative stability of the family members of quasi incompressible collision kernels for the navier stokes equations in confined flows was analyzed the linear stability analysis was simplified by assuming a uniform velocity level over the whole domain where only the wave numbers of the first harmonic normal to the flow direction were permitted a singular equilibrium function that maximizes the critical velocity level was identified which was afterwards tested in particular cases of confined flows of interest validating the resulting procedure

fluid mechanics of the interface region between two porous layers flow through and over a fluid saturated porous layer is investigated the flow through a porous channel which is assumed to be governed by forchheimer equation is terminated by a porous layer possessing a different structure the flow through which is governed by the brinkman equation at the interface between the physical regions matching conditions on the velocity and shear stress are imposed the flow through this configuration admits solutions which are linear combinations of polynomial and exponential functions the effect of the reynolds number and the darcy numbers on the interface velocity is presented in this work

leading off diagonal approximation for the spectral form factor for uniformly hyperbolic systems we consider the semiclassical approximation to the spectral form factor k for two dimensional uniformly hyperbolic systems with time reversal symmetry and derive the first off diagonal correction for small the result agrees with the 2 term of the form factor for the goe random matrix ensemble

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

a new iterative technique for solving nonlinear second order multi point boundary value problems abstract we present a semi analytical iterative method for solving nonlinear second order multi point boundary value problems to demonstrate the working of the method we consider a particular example of this class of problems in this example we demonstrate the accuracy and convergence of the method to the solution we demonstrate clearly that the method is accurate fast and has a reasonable order of convergence

a fractional step thermal lattice boltzmann model for high peclet number flow as a promising approach in hydrodynamics and thermodynamics modeling the lattice boltzmann method lbm still suffers severe numerical instability when the temperature field of the flow is convection dominant high peclet number despite a lot of research devoted to solve this problem worldwide to simulate high peclet number thermal flow at comparably few computational cost is still a hard work making it inefficient in practical use in this paper we combine the lbm and the fractional step method to propose a novel and stable thermal lattice boltzmann scheme for high peclet number flow without refining the lattice by numerical tests of thermal poiseuille flow and couette flow we quantify second order accuracy of the proposed model and through several cases of peclet number from low to high the superior stability and efficiency compared with existing thermal lattice boltzmann model

form factor for a family of quantum graphs an expansion to third order for certain types of quantum graphs we show that the random matrix form factor can be recovered to at least third order in the scaled time from periodic orbit theory we consider the contributions from pairs of periodic orbits represented by diagrams with up to two self intersections connected by up to four arcs and explain why all other diagrams are expected to give higher order corrections only for a large family of graphs with ergodic classical dynamics the diagrams that exist in the absence of time reversal symmetry sum to zero the mechanism for this cancellation is rather general which suggests that it also applies at higher orders in the expansion this expectation is in full agreement with the fact that in this case the linear contribution the diagonal approximation already reproduces the random matrix form factor for 1 for systems with time reversal symmetry there are more diagrams which contribute at third order we sum these contributions for quantum graphs with uniformly hyperbolic dynamics obtaining 2 3 in agreement with random matrix theory as in the previous calculation of the leading order correction to the diagonal approximation we find that the third order contribution can be attributed to exceptional orbits representing the intersection of diagram classes

non negativity and stability analyses of lattice boltzmann method for advection diffusion equation stability is one of the main concerns in the lattice boltzmann method lbm the objectives of this study are to investigate the linear stability of the lattice boltzmann equation with the bhatnagar gross krook collision operator lbgk for the advection diffusion equation ade and to understand the relationship between the stability of the lbgk and non negativity of the equilibrium distribution functions edfs this study conducted linear stability analysis on the lbgk whose stability depends on the lattice peclet number the courant number the single relaxation time and the flow direction the von neumann analysis was applied to delineate the stability domains by systematically varying these parameters moreover the dimensionless edfs were analyzed to identify the non negative domains of the dimensionless edfs as a result this study obtained linear stability and non negativity domains for three different lattices with linear and second order edfs it was found that the second order edfs have larger stability and non negativity domains than the linear edfs and outperform linear edfs in terms of stability and numerical dispersion furthermore the non negativity of the edfs is a sufficient condition for linear stability and becomes a necessary condition when the relaxation time is very close to 0 5 the stability and non negativity domains provide useful information to guide the selection of dimensionless parameters to obtain stable lbm solutions we use mass transport problems to demonstrate the consistency between the theoretical findings and lbm solutions

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

the role of the kinetic parameter in the stability of two relaxation time advection diffusion lattice boltzmann schemes in general explicit numerical schemes are only conditionally stable a particularity of lattice boltzmann multiple relaxation time mrt schemes is the presence of free kinetic relaxation parameters they do not appear in the transport coefficients of the modelled second order macroscopic equations but they have an impact on the effective accuracy and stability of the algorithm the simplest uniform choice the well known bgk srt model is often inadequate and therefore a compromise in the complexity of the model is sought for this purpose the von neumann stability analysis is performed for the d1q3 two relaxation time trt advection diffusion model the extended optimal eotrt model which relates the two collision times such that the most stable scheme is set by a suitable choice of the equilibrium parameters equal for any peclet number is then developed this extends the very recently derived optimal subclass otrt to larger combinations of physical and kinetic collision rates next we provide the necessary and or sufficient stability limits on the eotrt subclass for a wide range of velocity sets with and without numerical diffusion and delineate the interesting choices of free equilibrium weights for the d2q9 and d3q15 models the bgk srt model is without advanced advection properties we prove for minimal stencil schemes d1q3 d2q5 and d3q7 that the non negativity of the equilibrium distribution is necessary for its stability in the advection dominated limit beyond the eotrt and bgk srt subclasses of the trt model blind choices of the ghost collision number may result in quite unstable schemes even for positive equilibrium however we find that the d1q3 stability curves govern the advection properties of the multi dimensional models and a fuller picture of the trt stability properties begins to emerge

simulation of buoyancy driven flows in a vertical cylinder using a simple lattice boltzmann model axisymmetric thermal flow is of fundamental interest and practical importance however the work to design suitable and efficient lattice boltzmann models on axisymmetric thermal flows is quite rare in order to bridge the gap a simple lattice boltzmann model for axisymmetric thermal flow is proposed in this paper in the present study we show how to transform the governing equation for temperate field in the cylindrical coordinate system to the pseudo cartesian representation in the same way as that for the flow field therefore the flow field and the temperature field both are solved by the two dimensional five speed d2q5 lattice boltzmann model the treatment of the geometrical forcing due to the coordinate transformation and the physical forcing due to the temperature field is simpler than that in all existing models thanks to its intrinsic features the present model is more efficient more stable and much simpler than the existing models in this paper several kinds of nontrivial thermal buoyancy driven flows in vertical cylinders which are of interest from the standpoint of both basic fluid dynamics and practical applications are simulated by the present model

multiple relaxation time lattice boltzmann model for multiphase flow the lattice boltzmann method has shown promise in simulating multiphase flows however when using the bhatnagar gross krook bgk collision operator and polynomial equilibria numerical stability problems have been shown to occur as the relaxation time is decreased some authors have suggested the use of multiple relaxation time mrt models in lieu of the bgk collision operator which employs a single relaxation time to enhance numerical stability in this paper a mrt lattice boltzmann model for multiphase flow is developed and evaluated for accuracy in several test problems including oscillating liquid cylinders and capillary waves it is shown that the mrt model is able to achieve numerically stable results at lower viscosities relative to the corresponding bgk model

an exact trace formula and zeta functions for an infinite quantum graph with a non standard weyl asymptotics we study a quantum hamiltonian that is given by the negative laplacian and an infinite chain of like potentials with strength 0 on the half line and which is equivalent to a one parameter family of laplacians on an infinite metric graph this graph consists of an infinite chain of edges with the metric structure defined by assigning an interval in 0 ln to each edge with length we show that the one parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each 0 and prove that the dirichlet laplacian is the limit of the one parameter family in the strong resolvent sense the spectrum of the resulting dirichlet quantum graph is also purely discrete the eigenvalues are given by n n2 with multiplicities d n where d n denotes the divisor function we can thus relate the spectral problem of this infinite quantum graph to dirichlet s famous divisor problem and infer the non standard weyl asymptotics for the eigenvalue counting function based on an exact trace formula the voronoi summation formula we derive explicit formulae for the trace of the wave group the heat kernel the resolvent and for various spectral zeta functions these results enable us to establish a well defined renormalized secular equation and a selberg like zeta function defined in terms of the classical periodic orbits of the graph for which we derive an exact functional equation and prove that the analogue of the riemann hypothesis is true

spectral correlations of individual quantum graphs we investigate the spectral properties of chaotic quantum graphs we demonstrate that the energy average over the spectrum of individual graphs can be traded for the functional average over a supersymmetric nonlinear model action this proves that spectral correlations of individual quantum graphs behave according to the predictions of wigner dyson random matrix theory we explore the stability of the universal random matrix behavior with regard to perturbations and discuss the crossover between different types of symmetries

stabilization of the lattice boltzmann method by the h theorem a numerical test for a one dimensional benchmark shock tube problem we implement the lattice boltzmann method based on the h theorem i karlin a ferrante and h c ottinger europhys lett 47 182 1999 results of simulation demonstrate significant improvement of stability as compared to realizations without explicit entropic estimations

gpu accelerated lattice boltzmann model for shallow water flow and mass transport a lattice boltzmann method lbm for solving the shallow water equations swes and the advection dispersion equation is developed and implemented on graphics processing unit gpu based architectures a generalized lattice boltzmann equation glbe with a multiple relaxation time mrt collision method is used to simulate shallow water flow a two relaxation time trt method with two speed of sound techniques is used to solve the advection dispersion equation the proposed lbm is implemented to an nvidia computing processor in a single gpu workstation gpu computing is performed using the jacket gpu engine for matlab and cuda in the numerical examples the mrt lbm model and the trt lbm model are verified and show excellent agreement to exact solutions the mrt outperforms the single relaxation time srt collision operator in terms of stability and accuracy when the srt parameter is close to the stability limit of 0 5 mass transport with velocity dependent dispersion in shallow water flow is simulated by combining the mrt lbm model and the trt lbm model gpu performance with cuda code shows an order of magnitude higher than matlab jacket code moreover the gpu parallel performance increases as the grid size increases the results indicate the promise of the gpu accelerated lbm for modeling mass transport phenomena in shallow water flows copyright 2010 john wiley sons ltd