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

asymptotic preserving implicit explicit runge kutta methods for nonlinear kinetic equations we discuss implicit explicit imex runge kutta methods which are particularly adapted to stiff kinetic equations of boltzmann type we consider both the case of easy invertible collision operators and the challenging case of boltzmann collision operators we give sufficient conditions in order that such methods are asymptotic preserving and asymptotically accurate their monotonicity properties are also studied in the case of the boltzmann operator the methods are based on the introduction of a penalization technique for the collision integral this reformulation of the collision operator permits us to construct penalized imex schemes which work uniformly for a wide range of relaxation times avoiding the expensive implicit resolution of the collision operator finally we show some numerical results which confirm the theoretical analysis

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

a maxwellian lower bound for solutions to the boltzmann equation we prove that the solution of the spatially homogeneous boltzmann equation is bounded pointwise from below by a maxwellian i e a function of the formc1 exp c2v2 this holds for any initial data with bounded mass energy and entropy and for any positive timet t0 the constantsc1 andc2 depend on the mass energy and entropy of the initial data and ont0 0 only

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

analysis of spectral methods for the homogeneous boltzmann equation the development of accurate and fast algorithms for the boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis recently several works were devoted to the derivation of spectrally accurate schemes for the boltzmann equation but very few of them were concerned with the stability analysis of the method in particular there was no result of stability except when the method is modified in order to enforce the posivity preservation which destroys the spectral accuracy in this paper we propose a new method to study the stability of homogeneous boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution this method takes advantage of the spreading property of the collision together with estimates on regularity and entropy production as an application we prove stability and convergence of spectral methods for the boltzmann equation when the discretization parameter is large enough with explicit bound

l estimates for the space homogeneous boltzmann equation this paper studies the boundedness of solutionsf of the initial value problem for the space homogeneous boltzmann equation for inverse kth power forces whenk 5 and under angular cutoff the main result is that if the initial value isf0 0 with 1 2 0 el1 and 1 sf0e l for somes 2 then 1 s ftel fort 0 and ess tsup 1 s f t 5

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

a fast iterative scheme for the linearized boltzmann equation iterative schemes to find steady state solutions to the boltzmann equation is efficient for highly rarefied gas flows but can be very slow to converge in the near continuum flow regime in this paper a synthetic iterative scheme is developed to speed up the solution of the linearized boltzmann equation by penalizing the collision operator l into the form l l n delta h n delta h where delta is the gas rarefaction parameter h is the velocity distribution function and n is a tuning parameter controlling the convergence rate the velocity distribution function is first solved by the conventional iterative scheme then it is corrected such that the macroscopic flow velocity is governed by a diffusion type equation that is asymptotic preserving into the navier stokes limit the efficiency of this new scheme is assessed by calculating the eigenvalue of the iteration as well as solving for poiseuille and thermal transpiration flows we find that the fastest convergence of our synthetic scheme for the linearized boltzmann equation is achieved when n delta is close to the average collision frequency the synthetic iterative scheme is significantly faster than the conventional iterative scheme in both the transition and the near continuum gas flow regimes moreover due to its asymptotic preserving properties the synthetic iterative scheme does not need high spatial resolution in the near continuum flow regime which makes it even faster than the conventional iterative scheme using this synthetic scheme with the fast spectral approximation of the linearized boltzmann collision operator poiseuille and thermal transpiration flows between two parallel plates through channels of circular rectangular cross sections and various porous media are calculated over the whole range of gas rarefaction finally the flow of a ne ar gas mixture is solved based on the linearized boltzmann equation with the lennard jones intermolecular potential for the first time and the difference between these results and those using the hard sphere potential is discussed

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

construction of discrete kinetic models with given invariants we consider the general problem of the construction of discrete kinetic models dkms with given conservation laws this problem was first stated by gatignol in connection with discrete models of the boltzmann equation be and it has been addressed in the last decade by several authors even though a practical criterion for the non existence of spurious conservation laws has been devised and a method for enlarging existing physical models by new velocity points without adding non physical invariants has been proposed a general algorithm for the construction of all normal physical discrete models with assigned conservation laws in any dimension and for any number of points is still lacking in the literature we introduce the most general class of discrete kinetic models and obtain a general method for the construction and classification of normal dkms in particular it is proved that for any given dimension d 2 and for any sufficiently large number n of velocities for example n 6 for the planar case d 2 there exists just a finite number of distinct classes of dkms we apply the general method in the particular cases of discrete velocity models dvms of the inelastic be and elastic be using our general approach to dkms and our results on normal dvms for a single gas we develop a method for the construction of the most natural from physical point of view subclass of normal dvms for binary gas mixtures we call such models supernormal models snms they have the property that by isolating the velocities of single gases involved in the mixture we also obtain normal dvms

spectral methods for the non cut off boltzmann equation and numerical grazing collision limit in this paper we study the numerical passage from the spatially homogeneous boltzmann equation without cut off to the fokker planck landau equation in the so called grazing collision limit to this aim we derive a fourier spectral method for the non cut off boltzmann equation in the spirit of 21 23 we show that the kernel modes that define the spectral method have the correct grazing collision limit providing a consistent spectral method for the limiting fokker planck landau equation in particular for small values of the scattering angle we derive an approximate formula for the kernel modes of the non cut off boltzmann equation which similarly to the fokker planck landau case can be computed with a fast algorithm the uniform spectral accuracy of the method with respect to the grazing collision parameter is also proved

unbounded derived categories and the finitistic dimension conjecture abstract we consider the question of whether the injective modules generate the unbounded derived category of a ring as a triangulated category with arbitrary coproducts we give an example of a non noetherian commutative ring where they don t but prove that they do for any noetherian commutative ring for non commutative finite dimensional algebras the question is open and we prove that if injectives generate for such an algebra then the finitistic dimension conjecture holds for that algebra

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

exponential runge kutta for the inhomogeneous boltzmann equations with high order of accuracy we consider the development of exponential methods for the robust time discretization of space inhomogeneous boltzmann equations in stiff regimes compared to the space homogeneous case or more in general to the case of splitting based methods studied in dimarco pareschi 7 a major difficulty is that the local maxwellian equilibrium state change with respect to time and thus needs a proper numerical treatment we show how to derive asymptotic preserving ap schemes of arbitrary order and in particular by using the shu osher representation of runge kutta methods we explore the monotonicity properties of such schemes like strong stability preserving ssp and positivity preserving several numerical results confirm our analysis

a consistency result for a discrete velocity model of the boltzmann equation in this paper we study the link between a certain class of discrete velocity models dvms and the boltzmann equation those models possess an infinite number of velocities laying on a regular grid of step h in r3 our aim is to prove that it is possible to construct models consistent with the boltzmann equation i e such that the discrete collision term can be seen as an approximation of the collision integral of the boltzmann equation

a numerical method for the accurate solution of the fokker planck landau equation in the nonhomogeneous case abstract a new approach for the accurate numerical solution of the fokker planck landau fpl equation in the nonhomogeneous case is presented the method couples through a time splitting algorithm a finite volume scheme for the transport with a fast spectral solver for the efficient solution of the collision operator recently introduced the scheme allows the use of different grids in the velocity space for the transport and the collision phases the use of a suitable explicit runge kutta solver for the time integration of the collision phase permits avoid once of excessive small time steps induced by the stiffness of the diffusive collision operator numerical results for both space homogeneous and space nonhomogeneous situations show the efficiency and the accuracy of the present method

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

cosupports and minimal pure injective resolutions of affine rings abstract we prove that any affine ring r over a field k has full cosupport i e the cosupport of r is equal to spec r using this fact we give a complete description of all terms in a minimal pure injective resolution of r provided that k 1 and dim r 2 or k 1 and dim r 2 as a corollary we obtain a partial answer to a conjecture by gruson

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

brownian approximation and monte carlo simulation of the non cutoff kac equation the non cutoff boltzmann equation can be simulated using the dsmc method by a truncation of the collision term however even for computing stationary solutions this may be very time consuming in particular in situations far from equilibrium by adding an appropriate diffusion to the dsmc method the rate of convergence when the truncation is removed may be greatly improved we illustrate the technique on a toy model the kac equation as well as on the full boltzmann equation in a special case

discrete velocity models of the boltzmann equation a survey on the mathematical aspects of the theory discrete velocity models of the boltzmann equation are of considerable conceptual interest in the kinetic theory of gases and at the same time a fascinating mathematical subject the last decade has seen a lot of work on this subject which is reviewed here we put an emphasis on mathematical aspects in particular the global existence question qualitative properties of solutions explicit solutions and the fluid dynamical limit the mechanical background is discussed more concisely in part i numerous contributions concerning other questions are outlined we try to give a complete reference list up to and including 1985

discrete velocity models for mixtures without nonphysical collision invariants an important aspect of constructing discrete velocity models dvms for the boltzmann equation is to obtain the right number of collision invariants it is a well known fact that dvms can also have extra collision invariants so called spurious collision invariants in plus to the physical ones a dvm with only physical collision invariants and so without spurious ones is called normal for binary mixtures also the concept of supernormal dvms was introduced meaning that in addition to the dvm being normal the restriction of the dvm to any single species also is normal here we introduce generalizations of this concept to dvms for multicomponent mixtures we also present some general algorithms for constructing such models and give some concrete examples of such constructions one of our main results is that for any given number of species and any given rational mass ratios we can construct a supernormal dvm the dvms are constructed in such a way that for half space problems as the milne and kramers problems but also nonlinear ones we obtain similar structures as for the classical discrete boltzmann equation for one species and therefore we can apply obtained results for the classical boltzmann equation

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

normal high order discrete velocity models of the boltzmann equation abstract this paper aims at approximations of the collision operator in the boltzmann equation the developed framework guarantees the normality of the approximation which means correct collision invariants h theorem and equilibrium solutions it fits into the discrete velocity model framework is given in such a way that it is understandable with undergraduate level mathematics and can be used to construct approximations with arbitrary high convergence orders at last we give an example alongside a numerical verification here the convergence orders range up to 3 2 and the time complexity is given by 3 1 2 4 2 3 in 2 3 dimensions

a stochastic galerkin method for the boltzmann equation with uncertainty we develop a stochastic galerkin method for the boltzmann equation with uncertainty the method is based on the generalized polynomial chaos gpc approximation in the stochastic galerkin framework and can handle random inputs from collision kernel initial data or boundary data we show that a simple singular value decomposition of gpc related coefficients combined with the fast fourier spectral method in velocity space allows one to compute the high dimensional collision operator very efficiently in the spatially homogeneous case we first prove that the analytical solution preserves the regularity of the initial data in the random space and then use it to establish the spectral accuracy of the proposed stochastic galerkin method several numerical examples are presented to illustrate the validity of the proposed scheme

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

cosupport computations for finitely generated modules over commutative noetherian rings abstract we show that the cosupport of a commutative noetherian ring is precisely the set of primes appearing in a minimal pure injective resolution of the ring as an application of this we prove that every countable commutative noetherian ring has full cosupport we also settle the comparison of cosupport and support of finitely generated modules over any commutative noetherian ring of finite krull dimension finally we give an example showing that the cosupport of a finitely generated module need not be a closed subset of spec r providing a negative answer to a question of sather wagstaff and wicklein 29

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

five dimensional su 2 agt conjecture and recursive formula of deformed gaiotto state this paper deals with the five dimensional pure su 2 alday gaiotto tachikawa agt conjecture proposed by awata and yamada j high energy phys 1001 125 2010 we give a conjecture on a recursive formula for the inner product of the deformed gaiotto state we also show that the k theoretic pure su 2 nekrasov partition function satisfies the same recursion relation therefore the five dimensional agt conjecture is reduced to our conjectural recursive formula

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

half space problems for a linearized discrete quantum kinetic equation we study typical half space problems of rarefied gas dynamics including the problems of milne and kramer for a general discrete model of a quantum kinetic equation for excitations in a bose gas in the discrete case the plane stationary quantum kinetic equation reduces to a system of ordinary differential equations these systems are studied close to equilibrium and are proved to have the same structure as corresponding systems for the discrete boltzmann equation then a classification of well posed half space problems for the homogeneous as well as the inhomogeneous linearized discrete kinetic equation can be made the number of additional conditions that need to be imposed for well posedness is given by some characteristic numbers these characteristic numbers are calculated for discrete models axially symmetric with respect to the x axis when the characteristic numbers change is found in the discrete as well as the continuous case as an illustration explicit solutions are found for a small sized model

numerical scheme for a spatially inhomogeneous matrix valued quantum boltzmann equation we develop an efficient algorithm for a spatially inhomogeneous matrix valued quantum boltzmann equation derived from the hubbard model the distribution functions are 2 i 2 matrix valued to accommodate the spin degree of freedom and the scalar quantum boltzmann equation is recovered as a special case when all matrices are proportional to the identity we use fourier discretization and fast fourier transform to efficiently evaluate the collision kernel with spectral accuracy and numerically investigate periodic dirichlet and maxwell boundary conditions model simulations quantify the convergence to local and global thermal equilibrium

minimal semi flat cotorsion replacements and cosupport abstract over a commutative noetherian ring r of finite krull dimension we show that every complex of flat cotorsion r modules decomposes as a direct sum of a minimal complex and a contractible complex moreover we define the notion of a semi flat cotorsion complex as a special type of semi flat complex and provide functorial ways to construct a quasi isomorphism from a semi flat complex to a semi flat cotorsion complex consequently every r complex can be replaced by a minimal semi flat cotorsion complex in the derived category over r furthermore we describe structure of semi flat cotorsion replacements by which we recover classic theorems for finitistic dimensions in addition we improve some results on cosupport and give a cautionary example we also explain that semi flat cotorsion replacements always exist and can be used to describe the derived category over any associative ring

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

conformal symmetry in two dimensions an explicit recurrence formula for the conformal partial wave amplitude an explicit recurrence relation for the conformal block functions is presented this relation permits one to evaluate thex expansion of these functions order by order and appropriate for numerical calculations

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

on the monodromy problem for the four punctured sphere we consider the monodromy problem for the four punctured sphere in which the character of one composite monodromy is fixed by looking at the expansion of the accessory parameter in the modulus x directly without taking the limit of the quantum conformal blocks for an infinite central charge the integrals that appear in the expansion of the volterra equation involve products of two hypergeometric functions to first order and up to four hypergeometric functions to second order it is shown that all such integrals can be computed analytically we give the complete analytical evaluation of the accessory parameter to first and second order in the modulus the results agree with the evaluation obtained by assuming the exponentiation hypothesis of the quantum conformal blocks in the limit of infinite central charge extension to higher orders is discussed

accurate numerical methods for the collisional motion of heated granular flows in this paper we extend the spectral method developed in l pareschi b perthame a fourier spectral method for homogeneous boltzmann equations trans theo stat phys 25 1996 369 383 l pareschi g russo numerical solution of the boltzmann equation i spectrally accurate approximation of the collision operator siam j numer anal 37 2000 1217 1245 to the case of the inelastic boltzmann equation describing the collisional motion of a granular gas with and without a heating source the schemes are based on a fourier representation of the equation in the velocity space and provide a very accurate description of the time evolution of the distribution function several numerical results in dimension one to three show the efficiency and accuracy of the proposed algorithms some mathematical and physical conjectures are also addressed with the aid of the numerical simulations

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

polology of superconformal blocks we systematically classify all possible poles of superconformal blocks as a function of the scaling dimension of intermediate operators for all superconformal algebras in dimensions three and higher this is done by working out the recently proven irreducibility criterion for parabolic verma modules for classical basic lie superalgebras the result applies to correlators for external operators of arbitrary spin and indicates presence of infinitely many short multiplets of superconformal algebras most of which are non unitary we find a set of poles whose positions are shifted by linear in mathcal n for mathcal n extended supersymmetry we find an interesting subtlety for 3d mathcal n extended superconformal algebra with mathcal n odd associated with odd non isotropic roots we also comment on further applications to superconformal blocks

approximation of the linearized boltzmann collision operator for hard sphere and inverse power law models a sequence of approximate linear collision models for hard sphere and inverse power law gases is introduced these models are obtained by expanding the linearized boltzmann collision operator into series and a practical algorithm is proposed for evaluating the coefficients in the series the sequence is proven to be convergent to the linearized boltzmann operator and it established a connection between the shakhov model and the linearized collision model the convergence is demonstrated by solving the spatially homogeneous boltzmann equation by observing the magnitudes of the coefficients simpler models are developed through removing small entries in the coefficient matrices

acceleration schemes of the discrete velocity method gaseous flows in rectangular microchannels the convergence rate of the discrete velocity method dvm which has been applied extensively in the area of rarefied gas dynamics is studied via a fourier stability analysis the spectral radius of the continuum form of the iteration map is found to be equal to one which justifies the slow convergence rate of the method next the efficiency of the dvm is improved by introducing various acceleration schemes the new synthetic type schemes speed up significantly the iterative convergence rate the spectral radius of the acceleration schemes is also studied and the so called h1 acceleration method is found to be the optimum one finally the two dimensional flow problem of a gas through a rectangular microchannel is solved using the new fast iterative dvm the number of required iterations and the overall computational time are significantly reduced providing experimental evidence of the analytic formulation the whole approach is demonstrated using the bgk and s kinetic models

a differential equation for a four point correlation function in liouville field theory and elliptic four point conformal blocks liouville field theory on a sphere is considered we explicitly derive a differential equation for four point correlation functions with one degenerate field we also introduce and study a class of four point conformal blocks which can be calculated exactly and represented by finite dimensional integrals of elliptic theta functions for an arbitrary intermediate dimension we also study the bootstrap equations for these conformal blocks and derive integral representations for corresponding four point correlation functions a relation between the one point correlation function of a primary field on a torus and a special four point correlation function on a sphere is proposed

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

regularity theory for the spatially homogeneous boltzmann equation with cut off we develop the regularity theory of the spatially homogeneous boltzmann equation with cut off and hard potentials for instance hard spheres by i revisiting the lp theory to obtain constructive bounds ii establishing propagation of smoothness and singularities iii obtaining estimates on the decay of the singularities of the initial datum our proofs are based on a detailed study of the regularity of the gain operator an application to the long time behavior is presented

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

existence stability and convergence of solutions of discrete velocity models to the boltzmann equation we prove the convergence of finite difference approximations to solutions of the boltzmann equation an essential step is the proof of convergence of discrete approximations to the collision integral this proof relies on our previous results on the consistency of this approximation for the space homogeneous problem we prove strong convergence of our discrete approximation to the strong solution of the boltzmann equation in the space dependent case we prove weak convergence to diperna lions solutions

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

global l p properties for the spatially homogeneous boltzmann equation this paper studies the lp behavior for 1 p of solutions of the nonlinear spatially homogeneous boltzmann equation for a class of collision kernels including inverse kth power forces with k 5 and angular cut off the following topics are treated differentiability in lp together with global boundedness in time for lp moments that exist initially translation continuity in lp uniformly in time and strong convergence to equilibrium