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

algorithms for two dimensional cutting stock and strip packing problems using dynamic programming and column generation abstract we investigate several two dimensional guillotine cutting stock problems and their variants in which orthogonal rotations are allowed we first present two dynamic programming based algorithms for the rectangular knapsack rk problem and its variants in which the patterns must be staged the first algorithm solves the recurrence formula proposed by beasley the second algorithm for staged patterns also uses a recurrence formula we show that if the items are not so small compared to the dimensions of the bin then these algorithms require polynomial time using these algorithms we solved all instances of the rk problem found at the or library including one for which no optimal solution was known we also consider the two dimensional cutting stock problem we present a column generation based algorithm for this problem that uses the first algorithm above mentioned to generate the columns we propose two strategies to tackle the residual instances we also investigate a variant of this problem where the bins have different sizes at last we study the two dimensional strip packing problem we also present a column generation based algorithm for this problem that uses the second algorithm above mentioned where staged patterns are imposed in this case we solve instances for two three and four staged patterns we report on some computational experiments with the various algorithms we propose in this paper the results indicate that these algorithms seem to be suitable for solving real world instances we give a detailed description a pseudo code of all the algorithms presented here so that the reader may easily implement these algorithms

two dimensional packing problems a survey we consider problems requiring to allocate a set of rectangular items to larger rectangular standardized units by minimizing the waste in two dimensional bin packing problems these units are finite rectangles and the objective is to pack all the items into the minimum number of units while in two dimensional strip packing problems there is a single standardized unit of given width and the objective is to pack all the items within the minimum height we discuss mathematical models and survey lower bounds classical approximation algorithms recent heuristic and metaheuristic methods and exact enumerative approaches the relevant special cases where the items have to be packed into rows forming levels are also discussed in detail

fast approximation schemes for two stage two dimensional bin packing we present an asymptotic fully polynomial time approximation scheme for the two dimensional generalization of bin packing which requires packing or cutting a given set of rectangles from the minimum number of square bins with the further restriction that packing the rectangles in the bins is done in two stages as is frequently the case in real world applications to the best of our knowledge this is the first approximation scheme for a nontrivial two dimensional and real world generalization of a classical one dimensional packing problem in which rectangles have to be packed in finite squares in addition to the approximability result we obtain as a byproduct of our analysis an interesting structural result namely the asymptotic optimality of the lower bound provided by the natural linear programming relaxation of the problem having one variable for each bin pattern that can be solved effectively by column generation techniques besides its practical relevance a main reason to concentrate on the two stage version of the problem is that its study continues to shed light on the general version in particular we strongly conjecture that the approximation guarantee of our algorithm for two dimensional bin packing without the two stage restriction is better than the best known ones a simplification of the method leads to an approximation scheme for the two stage version of two dimensional strip packing

nr xx simulation of microflows with shakhov model in this paper we propose a method to simulate the microflows with shakhov model using the nr xx method developed in z cai and r li siam j sci comput 32 2010 pp 2875 2907 z cai r li and y wang commun comput phys 11 2012 pp 1415 1438 z cai r li and y wang j sci comput to appear the equation under consideration is the boltzmann equation with force terms and the shakhov model is adopted to achieve the correct prandtl number as the focus of this paper we derive a uniform framework for different order moment systems on the wall boundary conditions which is a major difficulty in the moment methods numerical examples for both steady and unsteady problems are presented to show the convergence in the number of moments

a harmonic algorithm for the 3d strip packing problem in the three dimensional 3d strip packing problem we are given a set of 3d rectangular items and a 3d box b the goal is to pack all the items in b such that the height of the packing is minimized we consider the most basic version of the problem where the items must be packed with their edges parallel to the edges of b and cannot be rotated building upon caprara s work for the two dimensional 2d bin packing problem we obtain an algorithm that given any epsilon 0 achieves an approximation of t infty epsilon approx1 69103 epsilon where t infty is the well known number that occurs naturally in the context of bin packing our key idea is to establish a connection between bin packing solutions for an arbitrary instance i and the strip packing solutions for the corresponding instance obtained from i by applying the harmonic transformation to certain dimensions based on this connection we also give a simple alternate proof of the t infty epsilon approximation for 2d

affordable robust moment closures for cfd based on the maximum entropy hierarchy the use of moment closures for the prediction of continuum and moderately non equilibrium flows offers modelling and numerical advantages over other methods the maximum entropy hierarchy of moment closures holds the promise of robustly hyperbolic stable moment equations however their are two issues that limit their practical implementation firstly for closures that have a treatment for heat transfer fluxes cannot be written in closed form and a very expensive iterative procedure is required at every flux evaluation secondly for these same closures there are physically possible moment states for which the entropy maximization problem has no solution and the entire framework breaks down this paper demonstrates that affordable closed form moment closures that are inspired by the maximum entropy framework can be proposed it is known that closing fluxes in the maximum entropy hierarchy approach a singularity as the region of non solvability is approached this paper shows that far from a disadvantage this singularity allows for smooth and accurate prediction of shock wave structure even for high mach numbers the presence of unphysical sub shocks within shock profile predictions of traditional closures has long been regarded as an unfortunate limitation of the entire moment closure technique the realization that smooth shock profiles are in fact possible for moment methods with a moderate number of moments greatly increases the method s applicability to high speed flows in this paper a 5 moment system for a simple one dimensional gas and a 14 moment system for realistic gases are developed and examined numerical solution for shock waves at a variety of incoming flow mach numbers demonstrate both the robustness and the accuracy of the closures

faster and simpler approximation algorithms for mixed packing and covering problems we propose an algorithm for approximately solving the mixed packing and covering problem given a convex compact set 0 b r n either compute x b such that f x 1 e a and g x 1 e b or decide that x b f x a g x b 0 here f g b r m are vectors whose components are m non negative convex and concave functions respectively and a b r m are constant positive vectors our algorithm requires an efficient feasibility oracle or block solver which given vectors c d r m and a r computes x b such that c tf x d tg x a or correctly decides that no such x b exists our algorithm which is based on the lagrangian or price directive decomposition method generalizes the result from k jansen approximation algorithm for the mixed fractional packing and covering problem in proceedings of 3rd ifip conference on theoretical computer science ifip tcs 2004 kluwer 2004 pp 223 236 siam journal on optimization 17 2006 331 352 and needs only o m lnm e 2ln e 1 iterations or calls to the feasibility oracle furthermore we show that a more general block solver can be used to obtain a more general approximation within the same runtime bound

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

the global homological dimension of the ring of differential operators on a nonsingular variety over a field of positive characteristic let k be a commutative ring and a a commutative k algebra in section 2 we define d a the ring of k linear differential operators on a if k is a field of characteristic zero and a is the coordinate ring of a nonsingular affine algebraic variety over k then it is shown in 3 51 that the global homological dimension of d a gl dim d a equals the dimension of the variety here we prove that if k is an algebraically closed field of characteristic p 0 and a is the coordinate ring of a nonsingular afline algebraic variety x over k then gl dim d a dim x in s it is shown that the weak global dimension of d a w dim d a equals dim x as d a is not noetherian there is no apriori reason for w dim d a and gl dim d a to be equal it is a relatively straightforward matter to see that d a is a union of subalgebras each of which has global dimension equal to dim x and so a theorem of berstein 2 gives gl dim d a d dim x 1 so the point is to show that in this particular situation berstein s result can be improved to show gl dim d a dim x that the global dimension is bounded below by dim x is a consequence of the fact that w dim d a dim x

a robust numerical method for the r13 equations of rarefied gas dynamics application to lid driven cavity in this work we present a finite difference scheme to compute steady state solutions of the regularized 13 moment r13 equations of rarefied gas dynamics the scheme allows fast solutions for 2d and 3d boundary value problems bvps with velocity slip and temperature jump boundary conditions the scheme is applied to the lid driven cavity problem for knudsen numbers up to 0 7 the results compare well with those obtained from more costly solvers for rarefied gas dynamics such as the integro moment method imm and the direct simulation monte carlo dsmc method the r13 equations yield better results than the classical navier stokes fourier equations for this boundary value problem since they give an approximate description of knudsen boundary layers at moderate knudsen numbers the r13 based numerical solutions are computationally economical and may be considered as a reliable alternative mathematical model for complex industrial problems at moderate knudsen numbers

a new approximation method for set covering problems with applications to multidimensional bin packing in this paper we introduce a new general approximation method for set covering problems based on the combination of randomized rounding of the near optimal solution of the linear programming lp relaxation leading to a partial integer solution and the application of a well behaved approximation algorithm to complete this solution if the value of the solution returned by the latter can be bounded in a suitable way as is the case for the most relevant generalizations of bin packing the method leads to improved approximation guarantees along with a proof of tighter integrality gaps for the lp relaxation for d dimensional vector packing we obtain a polynomial time randomized algorithm with asymptotic approximation guarantee arbitrarily close to ln d 1 for d 2 this value is 1 693 dots i e we break the natural 2 barrier for this case moreover for small values of d this is a notable improvement over the previously known o ln d guarantee by chekuri and khanna siam j comput 33 2004 pp 837 851 for two dimensional bin packing with and without rotations we obtain polynomial time randomized algorithms with asymptotic approximation guarantee 1 525 dots improving upon previous algorithms with asymptotic performance guarantees arbitrarily close to 2 by jansen and van stee on strip packing with rotations in proceedings of the 37th annual acm symposium on the theory of computing 2005 pp 755 761 for the problem with rotations and 1 691 ldots by caprara math oper res 33 2008 pp 203 215 for the problem without rotations the previously unknown key property used in our proofs follows from a retrospective analysis of the implications of the landmark bin packing approximation scheme by fernandez de la vega and lueker combinatorica 1 1981 pp 349 355 we prove that their approximation scheme is subset oblivious which leads to numerous applications

a strip packing algorithm with absolute performance bound 2 this paper proposes a new approximation algorithm m for packing rectangles into a strip with unit width and unbounded height so as to minimize the total height of the packing it is shown that for any list l of rectangles m l leq 2 cdot mbox opt l where m l is the strip height actually used by the algorithm m when applied to l and opt l is the minimum possible height within which the rectangles in l can be packed

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

acceleration for microflow simulations of high order moment models by using lower order model correction we study the acceleration of steady state computation for microflow which is modeled by the high order moment models derived recently from the steady state boltzmann equation with bgk type collision term by using the lower order model correction a novel nonlinear multi level moment solver is developed numerical examples verify that the resulting solver improves the convergence significantly thus is able to accelerate the steady state computation greatly the behavior of the solver is also numerically investigated it is shown that the convergence rate increases indicating the solver would be more efficient as the total levels increases three order reduction strategies of the solver are considered numerical results show that the most efficient order reduction strategy would be ml1 ml 2

approximate strip packing in this paper we establish an algorithmic framework between bin packing and strip packing with which strip packing can be very well approximated by applying some bin packing algorithms more precisely we obtain the following results 1 any off line bin packing algorithm can be applied to strip packing maintaining almost the same asymptotic worst case ratio 2 a class of harmonic based algorithms for bin packing such as refined harmonic modified harmonic harmonic can be applied to online strip packing in particular we show that online strip packing admits an upper bound of 1 58889 on the asymptotic competitive ratio for any arbitrarily small 0 this significantly improves the previously best bound of 1 6910 and affirmatively answers an open question posed by csirik and woeginger 1997 moreover the time complexity mainly depends on a sorting procedure and the bin packing algorithms employed

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

the two dimensional cutting stock problem revisited in the strip packing problem a standard version of the two dimensional cutting stock problem the goal is to pack a given set of rectangles into a vertical strip of unit width so as to minimize the total height of the strip needed the k stage guillotine packings form a particularly simple and attractive family of feasible solutions for strip packing we present a complete analysis of the quality of k stage guillotine strip packings versus globally optimal packings k 2 stages cannot guarantee any bounded asymptotic performance ratio k 3 stages lead to asymptotic performance ratios arbitrarily close to 1 69103 this bound is tight finally k 4 stages yield asymptotic performance ratios arbitrarily close to 1

bin packing in multiple dimensions inapproximability results and approximation schemes we study the following packing problem given a collection of d dimensional rectangles of specified sizes pack them into the minimum number of unit cubes we show that unlike the one dimensional case the two dimensional packing problem cannot have an asymptotic polynomial time approximation scheme aptas unless pnp on the positive side we give an aptas for the special case of packing d dimensional cubes into the minimum number of unit cubes second we give a polynomial time algorithm for packing arbitrary two dimensional rectangles into at most opt square bins with sides of length 1 where opt denotes the minimum number of unit bins required to pack these rectangles interestingly this result has no additive constant term i e is not an asymptotic result as a corollary we obtain the first approximation scheme for the problem of placing a collection of rectangles in a minimum area encasing rectangle

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

general multiprocessor task scheduling approximate solutions in linear time we study the problem of scheduling n independent tasks on a set of m parallel processors where the execution time of a task is a function of the subset of processors assigned to the task for any fixed m we propose a fully polynomial approximation scheme that for any fixed epsilon 0 finds a preemptive schedule of length at most 1 epsilon times the optimum in o n time we also discuss the nonpreemptive variant of the problem and present for any fixed m a polynomial approximation scheme that computes an approximate solution of any fixed accuracy in linear time in terms of the running time this linear complexity bound gives a substantial improvement of the best previously known polynomial bound j chen and a miranda siam j comput 31 2001 pp 1 17

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

bass numbers of local cohomology modules let a be a regular local ring of positive characteristic this paper is concerned with the local cohomology modules of a itself but with respect to an arbitrary ideal of a the results include that all the bass numbers of all such local cohomology modules are finite that each such local cohomology module has finite set of associated prime ideals and that whenever such a local cohomology module is artinian then it must be injective this last result had been proved earlier by hartshorne and speiser under the additional assumptions that a is complete and contains its residue field which is perfect the paper ends with some low dimensional evidence related to questions about whether the analogous statements for regular local rings of characteristic 0 are true

an approximation algorithm for the general max min resource sharing problem we propose an approximation algorithm for the general max min resource sharing problem with m nonnegative concave constraints on a convex set b the algorithm is based on a lagrangian decomposition method and it uses a c approximation algorithm called approximate block solver for a simpler maximization problem over the convex set b we show that our algorithm achieves within iterations or calls to the approximate block solver a solution for the general max min resource sharing problem with approximation ratio the algorithm is faster and simpler than the previous known approximation algorithms for the problem 12 13

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

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

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

numerical regularized moment method of arbitrary order for boltzmann bgk equation we introduce a numerical method for solving grad s moment equations or regularized moment equations for an arbitrary order of moments in our algorithm we do not explicitly need the moment equations instead we directly start from the boltzmann equation and perform grad s moment method h grad commun pure appl math 2 1949 pp 331 407 and the regularization technique h struchtrup and m torrilhon phys fluids 15 2003 pp 2668 2680 numerically we define a conservative projection operator and propose a fast implementation which makes it convenient to add up two distributions and provides more efficient flux calculations compared with the classic method using explicit expressions of flux functions for the collision term the bgk model is adopted so that the production step can be done trivially based on the hermite expansion extensive numerical examples for one and two dimensional problems are presented convergence in moments can be validated by the numerical results for different numbers of moments

an efficient steady state solver for microflows with high order moment model in z hu r li and z qiao acceleration for microflow simulations of high order moment models by using lower order model correction j comput phys 327 225 244 2016 it has been successfully demonstrated that using lower order moment model correction is a promising idea to accelerate the steady state computation of high order moment models of the boltzmann equation to develop the existing solver the following aspects are studied in this paper first the finite volume method with linear reconstruction is employed for high resolution spatial discretization of the resulting equations second by introducing an appropriate parameter in the correction step it is found that the performance of the solver can be improved significantly i e more levels would be involved in the solver which further accelerates the convergence of the method third heun s method is employed as the smoother in each level to enhance the robustness of the solver numerical experiments in microflows are carried out to demonstrate the efficiency and to investigate the behavior of the new solver in addition several order reduction strategies for the choice of the order sequence of the solver are tested and the strategy ml 1 ml 2 is found to be most efficient

the intersection homology d module in finite characteristic for y a closed normal subvariety of codimension c of a smooth variety x brylinski and kashiwara showed that the local cohomology module c y x x contains a unique simple x submodule denoted by y x in this paper the analogous result is shown for x and y defined over a perfect field of finite characteristic moreover a local construction of y x is given relating it to the theory of tight closure from the construction one obtains a criterion for the x simplicity of c y x x

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

entropic quadrature for moment approximations of the boltzmann bgk equation abstract using the method of moments to approximate the solution of kinetic equations has become a standard technique especially in the context of solving the boltzmann equation to model rarefied gases one attractive example is the maximum entropy closure which however is computationally barely affordable in this paper we will combine the maximum entropy approach with the quadrature method of moments qmom two methods which have in some sense diametric properties thereby introducing the entropic quadrature eq closure eq calculates a sparse quadrature based reconstruction of the unknown velocity distribution like the qmom where the physical meaningful maximum entropy principle is employed as a criterion for selecting a quadrature among different quadratures fitting the given moments we will discuss the construction of the closure and give several numerical examples of its performance in one and two dimensions

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

model reduction of kinetic equations by operator projection by a further study of the mechanism of the hyperbolic regularization of the moment system for the boltzmann equation proposed in cai et al commun math sci 11 2 547 571 2013 we point out that the key point is treating the time and space derivative in the same way based on this understanding a uniform framework to derive globally hyperbolic moment systems from kinetic equations using an operator projection method is proposed the framework is so concise and clear that it can be treated as an algorithm with four inputs to derive hyperbolic moment systems by routine calculations almost all existing globally hyperbolic moment systems can be included in the framework as well as some new moment systems including globally hyperbolic regularized versions of grad s ordered moment systems and a multi dimensional extension of the quadrature based moment system

d module and f module length of local cohomology modules let r be a polynomial or power series ring over a field k we study the length of local cohomology modules hji r in the category of d modules and fmodules we show that the d module length of hji r is bounded by a polynomial in the degree of the generators of i in characteristic p 0 we obtain upper and lower bounds on the f module length in terms of the dimensions of frobenius stable parts and the number of special primes of local cohomology modules of r i the obtained upper bound is sharp if r i is an isolated singularity and the lower bound is sharp when r i is gorenstein and f pure we also give an example of a local cohomology module that has different d module and f module lengths

on discontinuous galerkin approximations of boltzmann moment systems with levermore closure abstract this work considers the discontinuous galerkin dg finite element discretization of first order systems of conservation laws derivable as moments of the kinetic boltzmann equation with levermore c d levermore moment closure hierarchies for kinetic theories j statist phys 83 5 6 1996 1021 1065 closure using standard energy analysis techniques a new class of energy stable numerical flux functions are devised for the dg discretization of boltzmann moment systems simplified energy stable numerical fluxes are then constructed which replace exact state space integration in the numerical flux with gauss lobatto quadrature numerical results for supersonic flow over a cylinder geometry in the continuum and transitional regimes using 5 and 10 moment approximations are presented using the newly devised dg discretizations

domain of definition of levermore s five moment system the simplest system in levermore s moment hierarchy involving moments higher than second order is the five moment closure it is obtained by taking velocity moments of the one dimensional boltzmann equation under the assumption that the velocity distribution is a maximum entropy function the moment vectors for which a maximum entropy function exists consequently make up the domain of definition of the system the aim of this article is a complete characterization of the structure of the domain of definition and the connected maximum entropy problem the space homogeneous case of the equation and numerical aspects are also addressed

globally hyperbolic regularization of grad s moment system in this paper we propose a globally hyperbolic regularization to the general grad s moment system in multidimensional spaces systems with moments up to an arbitrary order are studied the characteristic speeds of the regularized moment system can be analytically given and depend only on the macroscopic velocity and the temperature the structure of the eigenvalues and eigenvectors of the coefficient matrix is fully clarified the regularization together with the properties of the resulting moment systems is consistent with the simple one dimensional case discussed in 1 in addition all characteristic waves are proven to be genuinely nonlinear or linearly degenerate and the studies on the properties of rarefaction waves contact discontinuities and shock waves are included 2014 wiley periodicals inc

commitment under uncertainty two stage stochastic matching problems we define and study two versions of the bipartite matching problem in the framework of two stage stochastic optimization with recourse in one version the uncertainty is in the second stage costs of the edges and in the other version the uncertainty is in the set of vertices that needs to be matched we prove lower bounds and analyze efficient strategies for both cases these problems model real life stochastic integral planning problems such as commodity trading reservation systems and scheduling under uncertainty

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

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

a near optimal solution to a two dimensional cutting stock problem we present an asymptotic fully polynomial approximation scheme for strip packing or packing rectangles into a rectangle of fixed width and minimum height a classicalnp hard cutting stock problem the algorithm based on a new linear programming relaxation finds a packing ofn rectangles whose total height is within a factor of 1 e of optimal up to an additive term and has running time polynomial both in n and in 1 e

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 new lower bound for online strip packing in this paper we consider the online strip packing problem in which a list of online rectangles has to be packed without overlap or rotation into a strip of width 1 and infinite length so as to minimize the required height of the packing we derive a new improved lower bound of 3 5 2 2 618 for the competitive ratio for this problem this result improves the best known lower bound of 2 589

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

parameterized complexity of strip packing and minimum volume packing abstract we study the parameterized complexity of minimum volume packing and strip packing in the two dimensional version the input consists of a set of rectangles s with integer side lengths in the minimum volume packing problem given a set of rectangles s and a number k the goal is to decide if the rectangles can be packed in a bounding box of volume at most k in the strip packing problem we are given a set of rectangles s numbers w and k the objective is to find if all the rectangles can be packed in a box of dimensions w k we prove that the 2 dimensional volume packing is in fpt by giving an algorithm that runs in 2 2 k k o 1 time we also show that strip packing is w 1 hard even in two dimensions and give an fpt algorithm for a special case of strip packing some of our results hold for the problems defined in higher dimensions as well

rarefied flow computations using nonlinear model boltzmann equations high resolution finite difference schemes for solving the nonlinear model boltzmann equations are presented for the computations of rarefied gas flows the discrete ordinate method is first applied to remove the velocity space dependency of the distribution function which renders the model boltzmann equation in phase space to a set of hyperbolic conservation laws with source terms in physical space then a high order essentially nonoscillatory method due to harten et al j comput phys 71 231 1987 is adapted and extended to solve them explicit methods using operator splitting and implicit methods using the lower upper factorization are described to treat multidimensional problems the methods are tested for both steady and unsteady rarefied gas flows to illustrate its potential use the computed results using model boltzmann equations are found to compare well both with those using the direct simulation monte carlo results in the transitional regime flows and those with the continuum navier stokes calculations in near continuum regime flows

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

moment closure hierarchies for kinetic theories this paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory the first member of the hierarchy is the euler system which is based on maxwellian velocity distributions while the second member is based on nonisotropic gaussian velocity distributions the closure proceeds in two steps the first ensures that every member of the hierarchy is hyperbolic has an entropy and formally recovers the euler limit the second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct navier stokes behavior this is achieved through the introduction of a generalization of the bgk collision operator the simplest such system in three spatial dimensions is a 14 moment closure which also recovers the behavior of the grad 13 moment system when the velocity distributions lie near local maxwellians the closure procedure can be applied to a general class of kinetic theories

moment closure approximations of the boltzmann equation based on varphi divergences this paper is concerned with approximations of the boltzmann equation based on the method of moments we propose a generalization of the setting of the moment closure problem from relative entropy to varphi divergences and a corresponding closure procedure based on minimization of varphi divergences the proposed description encapsulates as special cases grad s classical closure based on expansion in hermite polynomials and levermore s entropy based closure we establish that the generalization to divergence based closures enables the construction of extended thermodynamic theories that avoid essential limitations of the standard moment closure formulations such as inadmissibility of the approximate phase space distribution potential loss of hyperbolicity and singularity of flux functions at local equilibrium the divergence based closure leads to a hierarchy of tractable symmetric hyperbolic systems that retain the fundamental structural properties of the boltzmann equation

strip packing with precedence constraints and strip packing with release times the strip packing problem seeks to tightly pack a set of n rectangles into a strip of fixed width and arbitrary height the rectangles model tasks and the height models time this paper examines two variants of strip packing when the rectangles to be placed have precedence constraints and when the rectangles have release times strip packing is used to model scheduling problems in which tasks require a contiguous subset of identical resources that are arranged in a linear topology the variants studied here are motivated by scheduling tasks for dynamically reconfigurable field programmable gate arrays fpgas comprised of a linear arrangement of k homogeneous computing resources where k is a fixed positive integer and each task occupies a contiguous subset of these resources for the case in which tasks have precedence constraints we give an o logn approximation algorithm we then consider the special case in which all the rectangles have uniform height and reduce it to the resource constrained scheduling studied by garey graham johnson and yao thereby extending their asymptotic results to our special case problem we also give an absolute 3 approximation for this special case problem for strip packing with release times we provide an asymptotic polynomial time approximation scheme we make the standard assumption that the rectangles have height at most 1 in addition we also require widths to be in 1k 1 for the fpga application this would imply that the rectangles are at least as wide as a column our running time is polynomial in n and 1 e but exponential in k

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

packing d dimensional bins in d stages we consider the d dimensional bin packing problem the most relevant generalization of classical bin packing and show a general result about the asymptotic worst case ratio of a wide class of approximation algorithms that construct solutions in d stages containing many heuristics previously considered in the literature moreover we give the exact value of the asymptotic worst case ratio between the optimal solution and the best solution obtainable by packing the items in d stages showing how to achieve such a ratio efficiently the key property behind our result is the asymptotic optimality of the fractional relaxation of one dimensional bin packing an important by product of the approximation schemes for the problem from the 1980s which to the best of our knowledge is used here for the first time only for the sake of the analysis for the two dimensional case we push the approximablity threshold below 2 after more than 20 years reaching 1 691 but also improve and widely simplify the analysis of the previous best method from the 1980s moreover for d 3 we improve the previous approximability threshold by a factor of 1 691 a notable improvement when d is small

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

combinatorial benders cuts for the strip packing problem we study the strip packing problem in which a set of two dimensional rectangular items has to be packed in a rectangular strip of fixed width and infinite height with the aim of minimizing the height used the problem is important because it models a large number of real world applications including cutting operations where stocks of materials such as paper or wood come in large rolls and have to be cut with minimum waste scheduling problems in which tasks require a contiguous subset of identical resources and container loading problems arising in the transportation of items that cannot be stacked one over the other the strip packing problem has been attacked in the literature with several heuristic and exact algorithms nevertheless benchmark instances of small size remain unsolved to proven optimality in this paper we propose a new exact method that solves a large number of the open benchmark instances within a limited computational effort our method is based on a benders decomposition in whi

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

finiteness properties of local cohomology modules an application of d modules to commutative algebra the main goal of this paper is to establish finiteness properties of local cohomology modules in characteristic 0 that would be analogous to those proven by c huneke and r sharp in characteristicp 0 our method based on the theory of algebraicd modules seems to be the first application ofd modules to commutative algebra

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

a computational strategy for the regularized 13 moment equations with enhanced wall boundary conditions the challenge of modeling low speed rarefied gas flow in the transition regime is well known in this paper we propose a numerical solution procedure for the regularized 13 moment equations within a finite volume framework the stress and heat flux equations arising in the method of moments are transformed into the governing equations for the stress and heat flux deviators based on their first order approximation to model confined flows a complete set of wall boundary conditions for the 13 moment equations are derived based on the maxwell wall boundary model this has been achieved by expanding the molecular distribution function to fourth order accuracy in hermite polynomials empirical correction factors are introduced into the boundary conditions and calibrated against direct simulation monte carlo data the numerical predictions obtained from the regularized 13 moment equations and the navier stokes fourier equations are compared with data generated using the direct simulation monte carlo method for planar couette flow for a range of wall velocities and knudsen numbers 0 012 1 0 the results indicate that the regularized 13 moment equations are in good qualitative agreement with the direct simulation monte carlo data the results also highlight limitations that are caused by the use of a first order expansion of the third moment

fast approximation algorithms for fractional packing and covering problems this paper presents fast algorithms that find approximate solutions for a general class of problems which we call fractional packing and covering problems the only previously known algorithms for solving these problems are based on general linear programming techniques the techniques developed in this paper greatly outperform the general methods in many applications and are extensions of a method previously applied to find approximate solutions to multicommodity flow problems our algorithm is a lagrangian relaxation technique an important aspect of our results is that we obtain a theoretical analysis of the running time of a lagrangian relaxation based algorithm we give several applications of our algorithms the new approach yields several orders of magnitude of improvement over the best previously known running times for algorithms for the scheduling of unrelated parallel machines in both the preemptive and the nonpreemptive models for the job shop problem for the held and karp bound for the tr

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

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

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

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

performance bounds for orthogonal oriented two dimensional packing algorithms two orthogonal oriented two dimensional packing algorithms are presented and their behavior is analyzed under several assumptions for one of the algorithms we show that the ratio of the height used by the algorithm to the optimal height is asymptotically bounded by frac 4 3 this bound is an improvement over similar bounds for previously proposed algorithms

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

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

the d module structure of r f modules let r be a regular ring essentially of finite type over a perfect field k an r module m is called a unit r f module if it comes equipped with an isomorphism f e m m where f denotes the frobenius map on spec r and f e is the associated pullback functor it is well known that m then carries a natural d r module structure in this paper we investigate the relation between the unit r f structure and the induced d r structure on m in particular it is shown that if k is algebraically closed and m is a simple finitely generated unit r f module then it is also simple as a d r module an example showing the necessity of k being algebraically closed is also given

the gaussian moment closure for gas dynamics the moment closure method of levermore applied to the boltzmann equation for rarefied gas dynamics leads to a hierarchy of symmetric hyperbolic systems of partial differential equations the euler system is the first member of this hierarchy of closures in this paper we investigate the next member the 10 moment gaussian closure we first reduce the collision term to an integral which may be explicitly evaluated for the special case of maxwell molecular interaction the resulting collision term for this case is shown to be equivalent to the term obtained by replacing the boltzmann collision operator with the bhatnagar gross and krook bgk approximation we then analyze the gaussian system applied to the canonical flow problem of a stationary planar shock an analytic shock profile for the gaussian closure is derived and compared with the numerical solutions of the boltzmann and navier stokes equations the results show reasonable agreement for weak shocks and close agreement between the downstream ga

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

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

improved approximation for two dimensional strip packing with polynomial bounded width abstract we study the well known two dimensional strip packing problem given a set of rectangular axis parallel items and a strip of width w with infinite height the objective is to find a packing of all items into the strip which minimizes the packing height lately it has been shown that the lower bound of 3 2 of the absolute approximation ratio can be beaten when we allow a pseudo polynomial running time of type n w f 1 e if w is polynomially bounded by the number of items this is a polynomial running time the currently best pseudo polynomial approximation algorithm by nadiradze and wiese achieves an approximation ratio of 1 4 e we present a pseudo polynomial algorithm with improved approximation ratio 4 3 e furthermore the presented algorithm has a significantly smaller running time as the 1 4 e approximation algorithm

boundary conditions for regularized 13 moment equations for micro channel flows boundary conditions are the major obstacle in simulations based on advanced continuum models of rarefied and micro flows of gases in this paper we present a theory how to combine the regularized 13 moment equations derived from boltzmann s equation with boundary conditions obtained from maxwell s kinetic accommodation model while for the linear case these kinetic boundary conditions suffice we need additional conditions in the non linear case these are provided by the bulk solutions obtained after properly transforming the equations while keeping their asymptotic accuracy with respect to boltzmann s equation after finding a suitable set of boundary conditions and equations a numerical method for generic shear flow problems is formulated several test simulations demonstrate the stable and oscillation free performance of the new approach

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

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

comparison of kinetic theory and hydrodynamics for poiseuille flow comparison of particle dsmc simulation with the numerical solution of the navier stokes ns equations for pressure driven plane poiseuille flow is presented and contrasted with that of the acceleration driven poiseuille flow although for the acceleration driven case dsmc measurements are qualitatively different from the ns solution at relatively low knudsen number the two are in somewhat better agreement for pressure driven flow