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

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

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

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

quantum algebra deforming maps clebsch gordan coefficients coproducts r and u matrices quantum algebra deforming maps explicitly define comultiplications that differ from the usual noncocommutative coproducts map induced coproducts are connected to the usual ones by similarity transformations u that may be expressed either in terms of clebsch gordan coefficients or in a universal operator form the product of two such u matrices yields the r matrix for a fixed value of the spectral parameter which bears on the yang baxterization of u as well as r all this is explicitly illustrated for the tensor product 1 2 j of su 2 q using several deforming maps whose coproducts are continuously connected by similarity transformations to form a two parameter manifold some observations are made on the general structure of u and r matrices and of coproduct manifolds based on the solutions of hierarchies of partial difference equations applications of deforming maps and u matrices to the physics of spin chains are outlined

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

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

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

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

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

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

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

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

on the symmetric square unstable local transfer we prove the fundamental lemma for spherical functions with respect to the natural induction lifting frompgl 2 topgl 3 which appears as the unstable counterpart of the stable symmetric square lifting fromsl 2 topgl 3 see iv for an introduction to this project and vi for the final results thus spherical functions onpgl 2 andpgl 3 which correspond to each other by satisfying an elementary representation theoretic relation are shown to have matching orbital integrals the proof of this local statement is based on an application of the global trace formula

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

arbitrary lagrangian eulerian unstructured finite volume lattice boltzmann method for computing two dimensional compressible inviscid flows over moving bodies the objective of this study is to develop and apply an arbitrary lagrangian eulerian unstructured finite volume lattice boltzmann method ale fvlbm for solving two dimensional compressible inviscid flows around moving bodies the two dimensional compressible form of the lb equation is considered and the resulting lb equation is formulated in the ale framework on an unstructured body fitted mesh to correctly model the body shape and properly incorporate the mesh movement due to the body motion the spatial discretization of the resulting system of equations is performed by a second order cell centered finite volume method on arbitrary quadrilateral meshes and an implicit dual time stepping method is utilized for the time integration to stabilize the numerical solution appropriate numerical dissipation terms are added to the formulation at first the shock tube problem is computed to examine the accuracy of the solution obtained by applying the proposed fvlbm for this unsteady test case which includes shock expansion wave and contact discontinuity in the flow domain then the stationary isentropic vortex is simulated on both the stationary and moving meshes to assess the implementation of the geometric conservation law in enhancing the solution accuracy of the ale fvlbm the compressible inviscid flow in the transonic regime is then computed around the stationary naca0012 airfoil in order to further study the sensitivity of the solution method to the user defined parameters now the transonic inviscid flow is simulated over the pitching or plunging naca0012 airfoil to investigate the accuracy and capability of the proposed solution method ale fvlbm for the computation of the compressible flows over moving bodies finally the pitching or plunging naca0012 airfoil near the ground in the transonic inviscid flow is simulated as a practical and challenging problem to study the ground effect on the aerodynamic characteristics of the airfoil it is indicated that the solution methodology proposed based on the finite volume lbm formulated in the arbitrary lagrangian eulerian framework ale fvlbm is capable of accurately computing the compressible inviscid flows around the moving bodies with and without the ground effect

on the symmetric square twisted trace formula abstract we compute a trace formula for a test function on pgl 3 a twisted by the outer automorphism g j t g 1 j j 0 1 1 1 0 the resulting formula is then compared with trace formulae for h h 0 0 sl 2 and h 1 pgl 2 and matching functions f 0 and f 1 thereof we obtain a trace formula identity which plays a key role in our study of the symmetric square lifting from h a to g a our formulae are remarkably simple due to the introduction of a new concept of a regular function this eliminates the singular and weighted integrals in the trace formulae

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

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

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

asymptotic analysis of the lattice boltzmann equation in this article we analyze the lattice boltzmann equation lbe by using the asymptotic expansion technique we first relate the lbe to the finite discrete velocity model fdvm of the boltzmann equation with the diffusive scaling the analysis of this model directly leads to the incompressible navier stokes equations as opposed to the compressible navier stokes equations obtained by the chapman enskog analysis with convective scaling we also apply the asymptotic analysis directly to the fully discrete lbe as opposed to the usual practice of analyzing a continuous equation obtained through the taylor expansion of the lbe this leads to a consistency analysis which provides order by order information about the numerical solution of the lbe the asymptotic technique enables us to analyze the structure of the leading order errors and the accuracy of numerically derived quantities such as vorticity it also justifies the use of richardson s extrapolation method as an example a two dimensional taylor vortex flow is used to validate our analysis the numerical results agree very well with our analytic predictions

on the symmetric square orbital integrals j tg 1j for g in g the elements fi 8 of g f are called stably a conjugate if there is h in g f resp g with h a h1 conjugacy and stable conjugacy is defined analogously for h f and iii f on omitting a in sect 1 we show that the map n described in 1 5 below is a well defined bijection called the norm map from the set of stable a conjugacy classes in g f to the set of stable conjugacy classes in it f we also define a surjection n1 from the subset of this set described in 1 2 3 4 to the set of stable conjugacy classes in hi f in the case 4 n1 actually relates conjugacy classes let be an element of g f the set of eigenvalues of a is of the form 2 1 2 1 1 if 6a 1 put n 5 1 and ni 1 2 if 6a 6 is a non trivial unipotent let n6 and ni be the non trivial unipotent classes 3 if the eigenvalues 2 1 2 1 of 6a 6 are distinct let n6 be the class in h f determined by the eigenvalues 2 2 1 and ni6 the class in hi f with eigenvalues 4 1

on the symmetric square applications of a trace formula in this paper we prove the existence of the symmetric square lifting of admissible and of automorphic representations from the group sl 2 to the group pgl 3 complete local results are obtained relating the character of an sl 2 packet with the twisted character of self contragredient pgl 3 modules our global results relate packets of cuspidal representations of sl 2 with a square integrable component and self contragredient automorphic pgl 3 modules with a component coming from a square integrable one the sharp results which concern sl 2 rather than gl 2 are afforded by the usage of the trace formula the surjectivity and injectivity of the correspondence implies that any self contragredient automorphic pgl 3 module as above is a lift and that the space of cuspidal sl 2 modules with a square integrable component admits multiplicity one theorem and rigidity strong multiplicity one theorem for packets and not for individual representations

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

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

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

decomposition of tensor product representations of the unitary matrix quantum group suq 2 representations of suq 2 are labeled by a phase playing the role of a casimir operator the coproduct suq 2 is used to form direct product representations which can be reduced to irreducible ones direct product of two representations with given phases can be reduced to a representation with any given phase coefficients necessary for this procedure are calculated the calculation can also be performed in the fourier transformed space which replaces the phase by an integer

on the symmetric square definitions and lemmas we define the symmetric square lifting for admissible and automorphic representations from the group h h 0 sl 2 to the group g pgl 3 and derive its basic properties this lifting is defined by means of shintani character relations the definition is suggested by the computation of orbital integrals stable and unstable in our on the symmetric square orbital integrals math ann 279 1987 173 193 it is compatible with dual group homomorphisms 0 h g and 1 h 1 g where h 1 pgl 2 the lifting is proven for induced trivial and special representations and both spherical functions and orthogonality relations of characters are studied

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

representations and clebsch gordan coefficients for the jordanian quantum algebra representation theory for the jordanian quantum algebra is developed closed form expressions are given for the action of the generators of on the basis vectors of finite dimensional irreducible representations it is shown how representation theory of has a close connection to the combinatorial identities involving summation formulae a general formula is obtained for the clebsch gordan coefficients of these clebsch gordan coefficients are shown to coincide with those of su 2 for but for they are in general a non zero monomial in

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

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

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

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

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

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

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

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