spectral lagrangian methods for collisional models of non equilibrium statistical states we propose a new spectral lagrangian based deterministic solver for the non linear boltzmann transport equation bte in d dimensions for variable hard sphere vhs collision kernels with conservative or non conservative binary interactions the method is based on symmetries of the fourier transform of the collision integral where the complexity in its computation is reduced to a separate integral over the unit sphere s d 1 the conservation of moments is enforced by lagrangian constraints the resulting scheme implemented in free space is very versatile and adjusts in a very simple manner to several cases that involve energy dissipation due to local micro reversibility inelastic interactions or elastic models of slowing down process our simulations are benchmarked with available exact self similar solutions exact moment equations and analytical estimates for the homogeneous boltzmann equation both for elastic and inelastic vhs interactions benchmarking of the simulations involves the selection of a time self similar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for maxwell molecules as studied first in bobylev et al a v bobylev c cercignani g toscani proof of an asymptotic property of self similar solutions of the boltzmann equation for granular materials journal of statistical physics 111 2003 403 417 and generalized to a wide range of related models in bobylev et al a v bobylev c cercignani i m gamba on the self similar asymptotics for generalized non linear kinetic maxwell models communication in mathematical physics in press url the method also produces accurate results in the case of inelastic diffusive boltzmann equations for hard spheres inelastic collisions under thermal bath where overpopulated non gaussian exponential tails have been conjectured in computations by stochastic methods t v noije m ernst velocity distributions in homogeneously cooling and heated granular fluids granular matter 1 57 1998 m h ernst r brito scaling solutions of inelastic boltzmann equations with over populated high energy tails journal of statistical physics 109 2002 407 432 s j moon m d shattuck j swift velocity distributions and correlations in homogeneously heated granular media physical review e 64 2001 031303 i m gamba s rjasanow w wagner direct simulation of the uniformly heated granular boltzmann equation mathematical and computer modelling 42 2005 683 700 and rigorously proven in gamba et al i m gamba v panferov c villani on the boltzmann equation for diffusively excited granular media communications in mathematical physics 246 2004 503 541 39 and a v bobylev i m gamba v panferov moment inequalities and high energy tails for boltzmann equations with inelastic interactions journal of statistical physics 116 2004 1651 1682

a high order space momentum discontinuous galerkin method for the boltzmann equation in this paper we present a discontinuous galerkin method for the boltzmann equation the distribution function f is approximated by a shifted maxwellian times a polynomial in space and momentum while the test functions are chosen as polynomials the first property leads to consistency with the euler limit while the second property ensures conservation of mass momentum and energy the focus of the paper is on efficient algorithms for the boltzmann collision operator we transform between nodal hierarchical and polar polynomial bases to reduce the inner integral operator to diagonal form

a multiscale fast semi lagrangian method for rarefied gas dynamics in this paper we consider the extension of the method developed in dimarco and loubere 2013 22 23 with the aim of facing the numerical resolution of multi scale problems arising in rarefied gas dynamics the scope of this work is to consider situations in which the whole domain does not demand the use of a kinetic model everywhere this is the case of many realistic applications some regions of the computational domain require a microscopic description given by a kinetic model while the rest of the domain can be described by a coarser model of fluid type our aim is to show how the kinetic scheme developed in the pre cited articles is perfectly suited for building domain decomposition strategies which make the method more attractive with respect to classical numerical techniques for kinetic equations and multi scale realistic problems several numerical evidences are provided in this work in the two dimensional and three dimensional settings to assess the efficiency of the domain decomposition scheme

towards an ultra efficient kinetic scheme part i basics on the bgk equation in this paper we present a new ultra efficient numerical method for solving kinetic equations in this preliminary work we present the scheme in the case of the bgk relaxation operator the scheme being based on a splitting technique between transport and collision can be easily extended to other collisional operators as the boltzmann collision integral or to other kinetic equations such as the vlasov equation the key idea on which the method relies is to solve the collision part on a grid and then to solve exactly the transport linear part by following the characteristics backward in time the main difference between the method proposed and semi lagrangian methods is that here we do not need to reconstruct the distribution function at each time step this allows to tremendously reduce the computational cost of the method and it permits for the first time to the author s knowledge to compute solutions of full six dimensional kinetic equations on a single processor laptop machine numerical examples up to the full three dimensional case are presented which validate the method and assess its efficiency in 1d 2d and 3d

discrete velocity models and relaxation schemes for traffic flows we present simple discrete velocity models for traffic flows the novel feature in the corresponding relaxation system is the presence of nonnegative velocities only we show that in the small relaxation limit the discrete models reduce to the lighthill whitham richards equation in addition we propose second order schemes combined with implicit explicit time integrators as proper discretization of the relaxation type system numerical tests are carried out on various situations in traffic flow the results show that the proposed models are able to describe correctly the formation of backward waves induced by traffic jam

a convergence proof for bird s direct simulation monte carlo method for the boltzmann equation bird s direct simulation monte carlo method for the boltzmann equation is considered the limit as the number of particles tends to infinity of the random empirical measures associated with the bird algorithm is shown to be a deterministic measure valued function satisfying an equation close in a certain sense to the boltzmann equation a markov jump process is introduced which is related to bird s collision simulation procedure via a random time transformation convergence is established for the markov process and the random time transformation these results together with some general properties concerning the convergence of random measures make it possible to characterize the limiting behavior of the bird algorithm

spectral method for a kinetic swarming model in this paper we present the first numerical method for a kinetic description of the vicsek swarming model the kinetic model poses a unique challenge as there is a distribution dependent collision invariant to satisfy when computing the interaction term we use a spectral representation linked with a discrete constrained optimization to compute these interactions to test the numerical scheme we investigate the kinetic model at different scales and compare the solution with the microscopic and macroscopic descriptions of the vicsek model we observe that the kinetic model captures key features such as vortex formation and traveling waves

positivity preserving discontinuous galerkin schemes for linear vlasov boltzmann transport equations we develop a high order positivity preserving discontinuous galerkin dg scheme for linear vlasov boltzmann transport equations vlasov bte under the action of quadratically confined electrostatic potentials the solutions of such btes are positive probability distribution functions and it is very challenging to have a mass conservative high order accurate scheme that preserves positivity of the numerical solutions in high dimensions our work extends the maximum principle satisfying scheme for scalar conservation laws in a recent work by x zhang and c w shu to include the linear boltzmann collision term the dg schemes we developed conserve mass and preserve the positivity of the solution without sacrificing accuracy a discussion of the standard semi discrete dg schemes for the bte are included as a foundation for the stability and error estimates for this new scheme numerical results of the relaxation models are provided to validate the method

a direct method for the boltzmann equation based on a pseudo spectral velocity space discretization a deterministic method is proposed for solving the boltzmann equation the method employs a galerkin discretization of the velocity space and adopts as trial and test functions the collocation basis functions based on weights and roots of a gauss hermite quadrature this is defined by means of half and or full range hermite polynomials depending whether or not the distribution function presents a discontinuity in the velocity space the resulting semi discrete boltzmann equation is in the form of a system of hyperbolic partial differential equations whose solution can be obtained by standard numerical approaches the spectral rate of convergence of the results in the velocity space is shown by solving the spatially uniform homogeneous relaxation to equilibrium of maxwell molecules as an application the two dimensional cavity flow of a gas composed of hard sphere molecules is studied for different knudsen and mach numbers although computationally demanding the proposed method turns out to be an effective tool for studying subsonic slightly rarefied gas flows

tight linear programming relaxations of uncapacitated p hub median problems abstract the problem of locating hub facilities and allocating non hub nodes to those hubs arises frequently in the design of communication networks airline passenger flow and parcel delivery networks in this paper we consider uncapacitated multiple and single allocation p hub median problems we develop new mixed 0 1 linear formulations with tight linear programming relaxations the approach is tested on a well known and heavily used benchmark data set of real world problems with resulting lp relaxations ranging from 10010 to 391 250 variables and from 2 101 to 31 901 constraints which proved to be difficult linear programs yet this approach proved to be very effective in almost all instances the linear programming solution was integer in cases with fractional solutions the integrality was achieved by adding a small partial set of integrality constraints therefore we extended the range of optimally solvable instances of these np hard hub location problems which have defied researchers for the last ten years as an additional result for the single allocation case we were able to establish optimality of all heuristic solutions obtained via tabu search algorithm from a previous study for the more difficult single allocation p hub median problem we also used the best known heuristic solution as a guidance in adding integrality constraints this novel linkage between optimal and heuristic solutions has a potential impact in a number of other problem settings where efficient heuristic solutions exist and are probably but not provably optimal

self similarity and power like tails in nonconservative kinetic models in this paper we discuss the large time behavior of solution of a simple kinetic model of boltzmann maxwell type such that the temperature is time decreasing and or time increasing we show that under the combined effects of the nonlinearity and of the time monotonicity of the temperature the kinetic model has non trivial quasi stationary states with power law tails in order to do this we consider a suitable asymptotic limit of the model yielding a fokker planck equation for the distribution the same idea is applied to investigate the large time behavior of an elementary kinetic model of economy involving both exchanges between agents and increasing and or decreasing of the mean wealth in this last case the large time behavior of the solution shows a pareto power law tail numerical results confirm the previous analysis

exact solution of hub network design problems with profits abstract this paper studies hub network design problems with profits they consider a profit oriented objective that measure the tradeoff between the revenue due to served commodities and the overall network design and transportation costs an exact algorithmic framework is proposed for two variants of this class of problems where a sophisticated lagrangian function that exploits the structure of the problems is used to efficiently obtain bounds at the nodes of an enumeration tree in addition reduction tests and partial enumerations are used to considerably reduce the size of the problems and thus help decrease the computational effort numerical results on a set of benchmark instances with up to 100 nodes confirm the efficiency of the proposed algorithmic framework the proposed methodology can be used as a tool to solve more complex variants of this class of problems as well as other discrete location and network design problems involving servicing decisions

solving the boltzmann equation in n log 2 n in c mouhot and l pareschi fast algorithms for computing the boltzmann collision operator math comp to appear c mouhot and l pareschi c r math acad sci paris 339 2004 pp 71 76 fast deterministic algorithms based on spectral methods were derived for the boltzmann collision operator for a class of interactions including the hard spheres model in three dimensions these algorithms are implemented for the solution of the boltzmann equation in two and three dimensions first for homogeneous solutions then for general nonhomogeneous solutions the results are compared to explicit solutions when available and to monte carlo methods in particular the computational cost and accuracy are compared to those of monte carlo methods as well as to those of previous spectral methods finally for inhomogeneous solutions we take advantage of the great computational efficiency of the method to show an oscillation phenomenon of the entropy functional in the trend to equilibrium which was suggested in the work l desvillettes and c villani invent math 159 2005 pp 245 316

a quadratic integer program for the location of interacting hub facilities abstract this paper reports a new formulation of a general hub location model as a quadratic integer program non convexity of the objective function makes the problem difficult a variety of alternative solution strategies are discussed computational results from two simple heuristics are presented for the task of siting 2 3 or 4 hubs to serve interactions between sets of 10 15 20 and 25 u s cities the effects of different computational shortcuts are examined

on the parametrization of certain finite groups and their representations we discuss a convenient way of parametrizing certain finite groups they comprise groups with abelian normal subgroups and central extensions thereof they include the groups dn t o and their double groups as well as the su 3 subgroups tn 3n2 6n2 72 216 36 and their z3 extensions the method allows for a rapid calculation of representations and coupling coefficients in particular it solves the multiplicity problem of nonsimply reducible groups we illustrate the method treating the hessian group 216 and its central extension

lagrangian relaxation based approaches to capacitated hub and spoke network design problem abstract the main purpose of this paper is to introduce the capacitated hub and spoke network design problem in which hubs have limited capacity for channelling flows between the nodes served by the system the problem is formulated under a networking policy allowing both direct nonstop and hub connected one hub stop and two hub stop services between the nodes a branch and bound algorithm and a heuristic procedure partitioning the set of solutions on the basis of hub locations are presented in both algorithms after identifying a set of hub locations the problem is reduced into a smaller routing problem subgradient optimization is then applied to a lagrangian relaxation of the reduced problem the model and the proposed algorithms were applied to the air transportation system using data on passenger flows between the top forty us cities in 1989 accounting for 73 32 of the surveyed passenger flow computational results from 141 problems suggest that both algorithms are effective at finding good solutions to this problem

a fast algorithm for the energy space boson boltzmann collision operator abstract this paper introduces a fast algorithm for the energy space boson boltzmann collision operator compared to the direct o n3 calculation and the previous o n2 logn method markowich and pareschi 2005 the new algorithm runs in complexity o n log n which is optimal up to a logarithmic factor n is the number of grid points in energy space the basic idea is to partition the 3 d summation domain recursively into elementary shapes so that the summation within each shape becomes a special double convolution that can be computed efficiently by the fast fourier transform numerical examples are presented to illustrate the efficiency and accuracy of the proposed algorithm

galerkin petrov approach for the boltzmann equation abstract in this work we propose a new galerkin petrov method for the numerical solution of the classical spatially homogeneous boltzmann equation this method is based on an approximation of the distribution function by associated laguerre polynomials and spherical harmonics and test in a variational manner with globally defined three dimensional polynomials a numerical realisation of the algorithm is presented the algorithmic developments are illustrated with the help of several numerical tests

upper maxwellian bounds for the spatially homogeneous boltzmann equation for the spatially homogeneous boltzmann equation with cutoff hard potentials it is shown that solutions remain bounded from above uniformly in time by a maxwellian distribution provided the initial data have a maxwellian upper bound the main technique is based on a comparison principle that uses a certain dissipative property of the linear boltzmann equation implications of the technique to propagation of upper maxwellian bounds in the spatially inhomogeneous case are discussed

hierarchical boltzmann simulations and model error estimation abstract a hierarchical simulation approach for boltzmann s equation should provide a single numerical framework in which a coarse representation can be used to compute gas flows as accurately and efficiently as in computational fluid dynamics but a subsequent refinement allows to successively improve the result to the complete boltzmann result we use hermite discretization or moment equations for the steady linearized boltzmann equation for a proof of concept of such a framework all representations of the hierarchy are rotationally invariant and the numerical method is formulated on fully unstructured triangular and quadrilateral meshes using a implicit discontinuous galerkin formulation we demonstrate the performance of the numerical method on model problems which in particular highlights the relevance of stability of boundary conditions on curved domains the hierarchical nature of the method allows also to provide model error estimates by comparing subsequent representations we present various model errors for a flow through a curved channel with obstacles

integer programming formulations of discrete hub location problems abstract hubs are facilities that serve as transshipment and switching points for transportation and telecommunication systems with many origins and destinations this paper present integer programming formulations for four types of discrete hub location problems the p hub median problem the uncapacitated hub location problem p hub center problems and hub covering problems hub median and uncapacitated hub location problems have received limited attention from researchers this paper introduces discrete hub center and hub covering problems basic formulations and formulations with flow thresholds for spokes are presented

hub location under competition abstract hubs are consolidation and dissemination points in many to many flow networks hub location problem is to locate hubs among available nodes and allocate non hub nodes to these hubs the mainstream hub location studies focus on optimal decisions of one decision maker with respect to some objective s even though the markets that benefit hubbing are oligopolies therefore in this paper we propose a competitive hub location problem where the market is assumed to be a duopoly two decision makers or firms sequentially decide locations of their hubs and then customers choose one firm with respect to provided service levels each decision maker aims to maximize his her own market share we propose two problems for the leader former decision maker and follower latter decision maker r x p h u b m e d i a n o i d and r p h u b c e n t r o i d problems respectively both problems are proven to be np complete linear programming models are presented for these problems as well as exact solution algorithms for the r p h u b c e n t r o i d problem the performance of models and algorithms are tested by computational analysis conducted on cab and tr data sets

fast algorithms for computing the boltzmann collision operator the development of accurate and fast numerical schemes for the five fold boltzmann collision integral represents a challenging problem in scientific computing for a particular class of interactions including the so called hard spheres model in dimension three we are able to derive spectral methods that can be evaluated through fast algorithms these algorithms are based on a suitable representation and approximation of the collision operator explicit expressions for the errors in the schemes are given and spectral accuracy is proved parallelization properties and adaptivity of the algorithms are also discussed

a unified gas kinetic scheme for continuum and rarefied flows iv fluid dynamic equations are valid in their respective modeling scales such as the particle mean free path scale of the boltzmann equation and the hydrodynamic scale of the navier stokes ns equations with a variation of the modeling scales theoretically there should have a continuous spectrum of fluid dynamic equations even though the boltzmann equation is claimed to be valid in all scales many boltzmann solvers including direct simulation monte carlo method require the cell resolution to the order of particle mean free path scale therefore they are still single scale methods in order to study multiscale flow evolution efficiently the dynamics in the computational fluid has to be changed with the scales a direct modeling of flow physics with a changeable scale may become an appropriate approach the unified gas kinetic scheme ugks is a direct modeling method in the mesh size scale and its underlying flow physics depends on the resolution of the cell size relative to the particle mean free path the cell size of ugks is not limited by the particle mean free path with the variation of the ratio between the numerical cell size and local particle mean free path the ugks recovers the flow dynamics from the particle transport and collision in the kinetic scale to the wave propagation in the hydrodynamic scale the previous ugks is mostly constructed from the evolution solution of kinetic model equations even though the ugks is very accurate and effective in the low transition and continuum flow regimes with the time step being much larger than the particle mean free time it still has space to develop more accurate flow solver in the region where the time step is comparable with the local particle mean free time in such a scale there is dynamic difference from the full boltzmann collision term and the model equations this work is about the further development of the ugks with the implementation of the full boltzmann collision term in the region where it is needed the central ingredient of the ugks is the coupled treatment of particle transport and collision in the flux evaluation across a cell interface where a continuous flow dynamics from kinetic to hydrodynamic scales is modeled the newly developed ugks has the asymptotic preserving ap property of recovering the ns solutions in the continuum flow regime and the full boltzmann solution in the rarefied regime in the mostly unexplored transition regime the ugks itself provides a valuable tool for the non equilibrium flow study the mathematical properties of the scheme such as stability accuracy and the asymptotic preserving will be analyzed in this paper as well

product measure steady states of generalized zero range processes we establish necessary and sufficient conditions for the existence of factorizable steady states of the generalized zero range process on a periodic or infinite lattice this process allows transitions from a site i to a site i q involving a bounded number of multiple particles with rates depending on the content of the site i the direction q of movement and the number of particles moving we also show the sufficiency of a similar condition for the continuous time mass transport process where the mass at each site and the amount transferred in each transition are continuous variables we conjecture that this is also a necessary condition

numerical solution of the boltzmann equation i spectrally accurate approximation of the collision operator in this paper we show that the use of spectral galerkin methods for the approximation of the boltzmann equation in the velocity space permits one to obtain spectrally accurate numerical solutions at a reduced computational cost we prove that the spectral algorithm preserves the total mass and approximates with infinite order accuracy momentum and energy consistency of the method is also proved and a stability result for a smoothed positive scheme is given we demonstrate that the fourier coefficients associated with the collision kernel of the equation have a very simple structure and in some cases can be computed explicitly numerical examples for homogeneous test problems in two and three dimensions confirm the advantages of the method

towards an ultra efficient kinetic scheme part ii in a recent paper we presented a new ultra efficient numerical method for solving kinetic equations of the boltzmann type dimarco and loubere 2013 17 the key idea on which the method relies is to solve the collision part on a grid and then to solve exactly the transport part by following the characteristics backward in time on the contrary to classical semi lagrangian methods one does not need to reconstruct the distribution function at each time step this allows to tremendously reduce the computational cost and to perform efficient numerical simulations of kinetic equations up to the six dimensional case without parallelization however the main drawback of the method developed was the loss of spatial accuracy close to the fluid limit in the present work we modify the scheme in such a way that it is able to preserve the high order spatial accuracy for extremely rarefied and fluid regimes in particular in the fluid limit the method automatically degenerates into a high order method for the compressible euler equations numerical examples are presented which validate the method show the higher accuracy with respect to the previous approach and measure its efficiency with respect to well known schemes direct simulation monte carlo finite volume muscl weno

high order numerical methods for the space non homogeneous boltzmann equation in this paper we present accurate methods for the numerical solution of the boltzmann equation of rarefied gas the methods are based on a time splitting technique the transport is solved by a third order accurate in space positive and flux conservative pfc method the collision step is treated by a fourier approximation of the collision integral which guarantees spectral accuracy in velocity coupled with several high order integrators in time strang splitting is used to achieve second order accuracy in space and time several numerical tests illustrate the properties of the methods

relaxation schemes for nonlinear kinetic equations a class of numerical schemes for nonlinear kinetic equations of boltzmann type is described following wild s approach the solution is represented as a power series with parameter depending exponentially on the knudsen number this permits us to derive accurate and stable time discretizations for all ranges of the mean free path these schemes preserve the main physical properties positivity conservation of mass momentum and energy moreover for some particular models the entropy property is also shown to hold

convolution inequalities for the boltzmann collision operator we study integrability properties of a general version of the boltzmann collision operator for hard and soft potentials in n dimensions a reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution where the weight is given by an operator invariant under rotations using a symmetrization technique in l p we prove a young s inequality for hard potentials which is sharp for maxwell molecules in the l 2 case further we find a new hardy littlewood sobolev type of inequality for boltzmann collision integrals with soft potentials the same method extends to radially symmetric non increasing potentials that lie in some l s weak or l s the method we use resembles a brascamp lieb and luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting consequently it is closely connected to the classical analysis of young and hardy littlewood sobolev inequalities in all cases the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section in the spirit of grad cut off as an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions

slip boundary conditions for the compressible navier stokes equations the slip boundary conditions for the compressible navier stokes equations are derived systematically from the boltzmann equation on the basis of the chapman enskog solution of the boltzmann equation and the analysis of the knudsen layer adjacent to the boundary the resulting formulas of the slip boundary conditions are summarized with explicit values of the slip coefficients for hard sphere molecules as well as the bhatnagar gross krook model these formulas which can be applied to specific problems immediately help to prevent the use of often used slip boundary conditions that are either incorrect or without theoretical basis

finite subgroups of su 3 we present a new class of finite subgroups of su 3 of the form zm s zn semidirect product we also apply the methods used to investigate semidirect products to the known su 3 subgroups 3n2 and 6n2 and give analytic formulae for representations characters and clebsch gordan coefficients

discrete velocity models without nonphysical invariants models for mixtures of discrete velocity gases have been recently introduced by the authors and have produced unexpected results particularly with regard to the possible existence of nonphysical collision invariants here we discuss a method to construct models without spurious invariants the method can be extended to very general models including polyatomic gases chemical reactions etc

on the boltzmann equation for diffusively excited granular media we study the boltzmann equation for a space homogeneous gas of inelastic hard spheres with a diffusive term representing a random background forcing under the assumption that the initial datum is a nonnegative l 2 n function with bounded mass and kinetic energy second moment we prove the existence of a solution to this model which instantaneously becomes smooth and rapidly decaying under a weak additional assumption of bounded third moment the solution is shown to be unique we also establish the existence but not uniqueness of a stationary solution in addition we show that the high velocity tails of both the stationary and time dependent particle distribution functions are overpopulated with respect to the maxwellian distribution as conjectured by previous authors and we prove pointwise lower estimates for the solutions

representations and clebsch gordan coefficients of z metacyclic groups we investigate finite groups of the form zmszn and give all irreducible representations and clebsch gordan coefficients in analytic form two subclasses are considered which seem to be inportant for applications the m metacyclic groups which are important for spin systems and the k metacyclic groups which are the smallest finite groups that have an irreducible representation of dimension p 1 where p is prime

deterministic solution of the spatially homogeneous boltzmann equation using discontinuous galerkin discretizations in the velocity space abstract we present a new deterministic approach for the solution of the spatially homogeneous boltzmann kinetic equation based on nodal discontinuous galerkin dg discretizations in the velocity space in the new approach the collision operator has the form of a bilinear operator with a pre computed kernel its evaluation requires o n 5 operations at every point of the phase space where n is the number of degrees of freedom in one velocity dimension the method is generalized to any molecular potential results of numerical simulations are presented for the problem of spatially homogeneous relaxation for the hard spheres potential comparison with the method of direct simulation monte carlo showed excellent agreement

on the self similar asymptotics for generalized nonlinear kinetic maxwell models maxwell models for nonlinear kinetic equations have many applications in physics dynamics of granular gases economics etc in the present manuscript we consider such models from a very general point of view including those with arbitrary polynomial non linearities and in any dimension space it is shown that the whole class of generalized maxwell models satisfies properties one of which can be interpreted as an operator generalization of usual lipschitz conditions this property allows to describe in detail a behavior of solutions to the corresponding initial value problem in particular we prove in the most general case an existence of self similar solutions and study the convergence in the sense of probability measures of dynamically scaled solutions to the cauchy problem to those self similar solutions as time goes to infinity a new application of multi linear models to economics and social dynamics is discussed

a bi fidelity method for the multiscale boltzmann equation with random parameters abstract in this paper we study the multiscale boltzmann equation with multi dimensional random parameters by a bi fidelity stochastic collocation sc method developed in 52 70 71 by choosing the compressible euler system as the low fidelity model we adapt the bi fidelity sc method to combine computational efficiency of the low fidelity model with high accuracy of the high fidelity boltzmann model with only a small number of high fidelity asymptotic preserving solver runs for the boltzmann equation the bi fidelity approximation can capture well the macroscopic quantities of the solution to the boltzmann equation in the random space a priori estimate on the accuracy between the high and bi fidelity solutions together with a convergence analysis is established finally we present extensive numerical experiments to verify the efficiency and accuracy of our proposed method

exact and heuristic algorithms for the uncapacitated multiple allocation p hub median problem in this paper new milp formulations for the multiple allocation p hub median problem are presented these require fewer variables and constraints than those traditionally used in the literature an efficient heuristic algorithm based on shortest paths is described lp based solution methods as well as an explicit enumeration algorithm are developed to obtain exact solutions computational results are presented for well known problems from the literature which show that exact solutions can be found in a reasonable amount of computational time our algorithms are also benchmarked on a different data set this data set which includes problems that are larger than those used in the literature is based on a postal delivery network and has been treated by the authors in an earlier paper

discrete velocity models and numerical schemes for the boltzmann bgk equation in plane and axisymmetric geometries we present new numerical models for computing transitional or rarefied gas flows as described by the boltzmann bgk and bgk es equations we first propose a new discrete velocity model based on the entropy minimization principle this model satisfies the conservation laws and the entropy dissipation moreover the problem of conservation and entropy for axisymmetric flows is investigated we find algebraic relations that must be satisfied by the discretization of the velocity derivative appearing in the transport operator then we propose some models that satisfy these constraints owing to these properties we obtain numerical schemes that are economic in terms of discretization and robust in particular we develop a linearized implicit scheme for computing stationary solutions of the discrete velocity bgk and bgk es models this scheme is the basis of a code which can compute high altitude hypersonic flows in 2d plane and axisymmetric geometries our results are analyzed and compared to other methods