a hybridizable discontinuous galerkin method for stokes flow abstract in this paper we introduce a hybridizable discontinuous galerkin method for stokes flow the method is devised by using the discontinuous galerkin methodology to discretize a velocity pressure gradient formulation of the stokes system with appropriate choices of the numerical fluxes and by applying a hybridization technique to the resulting discretization one of the main features of this approach is that it reduces the globally coupled unknowns to the numerical trace of the velocity and the mean of the pressure on the element boundaries thereby leading to a significant reduction in the size of the resulting matrix moreover by using an augmented lagrangian method the globally coupled unknowns are further reduced to the numerical trace of the velocity only another important feature is that the approximations of the velocity pressure and gradient converge with the optimal order of k 1 in the l 2 norm when polynomials of degree k 0 are used to represent the approximate variables based on the optimal convergence of the hdg method we apply an element by element postprocessing scheme to obtain a new approximate velocity which converges with order k 2 in the l 2 norm for k 1 the postprocessing performed at the element level is less expensive than the solution procedure numerical results are provided to assess the performance of the method

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 hybridizable discontinuous galerkin method with fast convergence to steady state solutions of the gas kinetic equation abstract the mass flow rate of poiseuille flow of rarefied gas through long ducts of two dimensional cross sections with arbitrary shape is critical in the pore network modeling of gas transport in porous media here for the first time the high order hybridizable discontinuous galerkin hdg method is used to find the steady state solution of the linearized bhatnagar gross krook equation on two dimensional triangular meshes the velocity distribution function and its traces are approximated in piecewise polynomial spaces of degree up to 4 on the triangular meshes and mesh skeletons respectively by employing a numerical flux that is derived from the first order upwind scheme and imposing its continuity weakly on the mesh skeletons global systems for unknown traces are obtained with fewer coupled degrees of freedom when compared to the original discontinuous galerkin formulation to achieve fast convergence to the steady state solution a diffusion like equation for flow velocity which is asymptotic preserving into the fluid dynamic limit is solved by the hdg simultaneously on the same meshes the proposed hdg synthetic iterative scheme is proved to be accurate and efficient specifically for flows in the near continuum regime numerical simulations have shown that to achieve the same level of accuracy our scheme could be faster than the conventional iterative scheme by two orders of magnitude also it is faster than the synthetic iterative scheme based on the finite difference discretization in the spatial space by one order of magnitude in addition the implicit hdg method is more efficient than an explicit discontinuous galerkin gas kinetic solver as well as the implicit discontinuous galerkin scheme when the degree of approximating polynomial is larger than 2 the hdg synthetic iterative scheme is ready to be extended to simulate rarefied gas mixtures and the boltzmann collision operator

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

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

on the existence of solutions for a class of fourth order differential equations abstract we study the existence of the homoclinic solution of a class of nonlinear fourth order ordinary differential equations which involve particular cases of the extended fisher kolmogorov and swift hohenberg equations we address the case when the source term possesses a nonlinear dependence on the first order derivative of the solution the main result follows from an iterative method using the sequence of solutions obtained through the mountain pass technique in order to illustrate this method some numerical examples are shown

an implicit high order hybridizable discontinuous galerkin method for the incompressible navier stokes equations we present an implicit high order hybridizable discontinuous galerkin method for the steady state and time dependent incompressible navier stokes equations the method is devised by using the discontinuous galerkin discretization for a velocity gradient pressure velocity formulation of the incompressible navier stokes equations with a special choice of the numerical traces the method possesses several unique features which distinguish itself from other discontinuous galerkin methods first it reduces the globally coupled unknowns to the approximate trace of the velocity and the mean of the pressure on element boundaries thereby leading to a significant reduction in the degrees of freedom moreover if the augmented lagrangian method is used to solve the linearized system the globally coupled unknowns become the approximate trace of the velocity only second it provides for smooth viscous dominated problems approximations of the velocity pressure and velocity gradient which converge with the optimal order of k 1 in the l2 norm when polynomials of degree k 0 are used for all components of the approximate solution and third it displays superconvergence properties that allow us to use the above mentioned optimal convergence properties to define an element by element postprocessing scheme to compute a new and better approximate velocity indeed this new approximation is exactly divergence free h div conforming and converges with order k 2 for k 1 and with order 1 for k 0 in the l2 norm moreover a novel and systematic way is proposed for imposing boundary conditions for the stress viscous stress vorticity and pressure which are not naturally associated with the weak formulation of the method this can be done on different parts of the boundary and does not result in the degradation of the optimal order of convergence properties of the method extensive numerical results are presented to demonstrate the convergence and accuracy properties of the method for a wide range of reynolds numbers and for various polynomial degrees

semilinear elliptic variational inequalities with dependence on the gradient via mountain pass techniques abstract in this paper we consider a semilinear variational inequality with a gradient dependent nonlinear term obviously the nature of this problem is non variational nevertheless we study that problem associating a suitable semilinear variational inequality variational in nature with it and performing an iterative technique used in de figueiredo et al 2004 6 in order to treat semilinear elliptic equations when there is a gradient dependence on the nonlinearity we prove the existence of a non trivial non negative weak solution u for our problem using essentially variational methods a penalization technique and an iterative scheme via lewy stampacchia s estimates and regularity theory for elliptic equation we also show that u is differentiable and its gradient is h lder continuous on for any 0 1

existence of multiple solutions to nonlinear elliptic equations of nondivergence form abstract in this paper by means of a priori estimates and topological degree theory we prove the existence of two or more solutions to nonlinear elliptic equations of nondivergence form

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

existence of positive solutions for the p laplacian with p gradient term abstract in this paper we study the existence of positive solutions for the p laplacian involving a p gradient term due to the non variational structure and the fact that the nonlinearity may be critical or supercritical the variational method is no longer valid taking advantage of global c 1 estimates and the liouville type theorems we employ the blow up argument to obtain the a priori estimates on solutions and finally obtain the existence result based on the krasnoselskii fixed point theorem

discrete unified gas kinetic scheme for all knudsen number flows low speed isothermal case based on the boltzmann bgk bhatnagar gross krook equation in this paper a discrete unified gas kinetic scheme dugks is developed for low speed isothermal flows the dugks is a finite volume scheme with the discretization of particle velocity space after the introduction of two auxiliary distribution functions with the inclusion of collision effect the dugks becomes a fully explicit scheme for the update of distribution function furthermore the scheme is an asymptotic preserving method where the time step is only determined by the courant friedricks lewy condition in the continuum limit numerical results demonstrate that accurate solutions in both continuum and rarefied flow regimes can be obtained from the current dugks the comparison between the dugks and the well defined lattice boltzmann equation method d2q9 is presented as well

existence of solutions for a bi nonlocal fractional p kirchhoff type problem in this paper we are concerned with the existence of nonnegative solutions for a p kirchhoff type problem driven by a non local integro differential operator with homogeneous dirichlet boundary data as a particular case we study the following problem m x u s p p p s u f x u u s p p in u 0 in r n u s p p r 2 n u x u y p x y n p s d x d y where p s is a fractional p laplace operator is an open bounded subset of r n with lipschitz boundary m r 0 r is a continuous function and f r r 0 r is a continuous function satisfying the ambrosetti rabinowitz type condition the existence of nonnegative solutions is obtained by using the mountain pass theorem and an iterative scheme the main feature of this paper lies in the fact that the kirchhoff function m depends on x and the nonlinearity f depends on the energy of solutions

a gaussian quadrature procedure for use in the solution of the boltzmann equation and related problems abstract a new gaussian quadrature procedure is developed for integrals of the form 0 e y 2 y p y dy for p 0 1 and 2 recursion relations are derived for the coefficients in the general three term recurrence relation for the polynomials whose roots are the quadrature abscissae a comparison with the gauss laguerre quadrature procedure is presented solutions of the chemical kinetic boltzmann equation are obtained with a discrete ordinate method based on this gaussian quadrature procedure the results are compared with previous solutions obtained with a polynomial expansion method

discrete unified gas kinetic scheme for all knudsen number flows ii thermal compressible case this paper is a continuation of our work on the development of multiscale numerical scheme from low speed isothermal flow to compressible flows at high mach numbers in our earlier work z l guo et al phys rev e 88 033305 2013 a discrete unified gas kinetic scheme dugks was developed for low speed flows in which the mach number is small so that the flow is nearly incompressible in the current work we extend the scheme to compressible flows with the inclusion of thermal effect and shock discontinuity based on the gas kinetic shakhov model this method is an explicit finite volume scheme with the coupling of particle transport and collision in the flux evaluation at a cell interface as a result the time step of the method is not limited by the particle collision time with the variation of the ratio between the time step and particle collision time the scheme is an asymptotic preserving ap method where both the chapman enskog expansion for the navier stokes solution in the continuum regime and the free transport mechanism in the rarefied limit can be precisely recovered with a second order accuracy in both space and time the dugks is an idealized multiscale method for all knudsen number flow simulations a number of numerical tests including the shock structure problem the sod tube problem in a whole range of degree of rarefaction and the two dimensional riemann problem in both continuum and rarefied regimes are performed to validate the scheme comparisons with the results of direct simulation monte carlo dsmc and other benchmark data demonstrate that the dugks is a reliable and efficient method for multiscale flow problems

translationally invariant multipartite bell inequalities involving only two body correlators bell inequalities are natural tools that allow one to certify the presence of nonlocality in quantum systems the known constructions of multipartite bell inequalities contain however correlation functions involving all observers making their experimental implementation difficult the main purpose of this work is to explore the possibility of witnessing nonlocality in multipartite quantum states from the easiest to measure quantities that is the two body correlations in particular we determine all three and four partite bell inequalities constructed from one and two body expectation values that obey translational symmetry and show that they reveal nonlocality in multipartite states also by providing a particular example of a five partite bell inequality we show that nonlocality can be detected from two body correlators involving only nearest neighbours finally we demonstrate that any translationally invariant bell inequality can be maximally violated by a translationally invariant state and the same set of observables at all sites we provide a numerical algorithm allowing one to seek for maximal violation of a translationally invariant bell inequality this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

existence of solutions for fourth order elliptic equations of kirchhoff type abstract this paper is concerned with the existence of nontrivial solutions for a class of fourth order elliptic equations of kirchhoff type 1 2 u a b u 2 d x u f x u in u 0 u 0 on where a 0 b 0 are constants and 0 is a parameter we will show that there exists a such that 1 has nontrivial solutions for 0 by using the mountain pass techniques and the truncation method

periodic solutions for extended fisher kolmogorov and swift hohenberg equations by truncature techniques abstract combining truncature techniques with a variational approach we establish an existence result for nontrivial periodic solutions for a class of fourth order ordinary differential equations involving extended fisher kolmogorov and swift hohenberg equations

bounding the set of classical correlations of a many body system we present a method to certify the presence of bell correlations in experimentally observed statistics and to obtain new bell inequalities our approach is based on relaxing the conditions defining the set of correlations obeying a local hidden variable model yielding a convergent hierarchy of semidefinite programs sdp s because the size of these sdp s is independent of the number of parties involved this technique allows us to characterize correlations in many body systems as an example we illustrate our method with the experimental data presented in science 352 441 2016 scieas0036 807510 1126 science aad8665

stability for semilinear elliptic variational inequalities depending on the gradient abstract in this work we give a result concerning the continuous dependence on the data for weak solutions of a class of semilinear elliptic variational inequalities p n with a nonlinear term depending on the gradient of the solution this paper can be seen as the second part of the work matzeu and servadei 2010 9 in the sense that here we give a stability result for the c 1 weak solutions of problem p n found in matzeu and servadei 2010 9 through variational techniques to be precise we show that the solutions of p n found with the arguments of matzeu and servadei 2010 9 converge to a solution of the limiting problem p under suitable convergence assumptions on the data

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

a comparative study of an asymptotic preserving scheme and unified gas kinetic scheme in continuum flow limit asymptotic preserving ap schemes target to simulate both continuum and rarefied flows many existing ap schemes are capable of recovering the euler limit in the continuum regime however to get accurate navier stokes solutions is still challenging in order to distinguish physical mechanism underlining different ap schemes an implicit explicit imex ap method and the unified gas kinetic scheme ugks are employed to solve the bhatnagar gross krook bgk kinetic equation in both transition and continuum flow regimes as a benchmark test case the lid driven cavity flow is used for evaluating numerical performance of these two ap schemes the numerical results show that the ugks captures the viscous solution accurately the velocity profiles converge to the classical benchmark solutions in continuum regime with the mesh size being much larger than the local particle mean free path however the imex ap scheme seems to have difficulty to get these solutions in the corresponding limit the analysis demonstrates that the dissipation of ap schemes has to be properly controlled in the continuum flow regime through a delicate numerical treatment of collision and convection of the kinetic equation physically it becomes necessary to couple both the convection and collision terms in the flux evaluation in order to recover correct navier stokes limit

composite parameterization and haar measure for all unitary and special unitary groups we adopt the concept of the composite parameterization of the unitary group u d to the special unitary group su d furthermore we also consider the haar measure in terms of the introduced parameters we show that the well defined structure of the parameterization leads to a concise formula for the normalized haar measure on u d and su d with regard to possible applications of our results we consider the computation of high order integrals over unitary groups

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

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

a fast iterative model for discrete velocity calculations on triangular grids a fast synthetic type iterative model is proposed to speed up the slow convergence of discrete velocity algorithms for solving linear kinetic equations on triangular lattices the efficiency of the scheme is verified both theoretically by a discrete fourier stability analysis and computationally by solving a rarefied gas flow problem the stability analysis of the discrete kinetic equations yields the spectral radius of the typical and the proposed iterative algorithms and reveal the drastically improved performance of the latter one for any grid resolution this is the first time that stability analysis of the full discrete kinetic equations related to rarefied gas theory is formulated providing the detailed dependency of the iteration scheme on the discretization parameters in the phase space the corresponding characteristics of the model deduced by solving numerically the rarefied gas flow through a duct with triangular cross section are in complete agreement with the theoretical findings the proposed approach may open a way for fast computation of rarefied gas flows on complex geometries in the whole range of gas rarefaction including the hydrodynamic regime

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

exact solutions of the nonlinear boltzmann equation a review is given of research activities since 1976 on the nonlinear boltzmann equation and related equations of boltzmann type in which several rediscoveries have been made and several conjectures have been disproved subjects are i the bkw solution of the boltzmann equation for maxwell molecules first discovered by krupp in 1967 and the krook wu conjecture concerning the universal significance of the bkw solution for the large v t behavior of the velocity distribution functionf v t ii moment equations and polynomial expansions off v t iii model boltzmann equation for a spatially uniform system of very hard particles that can be solved in closed form for general initial conditions iv for maxwell and non maxwell type molecules there exist solutions of the nonlinear boltzmann equation with algebraic decrease at connections with nonuniqueness and violation of conservation laws v conjectured super h theorem and the bkw solution vi exactly soluble one dimensional boltzmann equation with spatial dependence

bell s theorem for general n qubit states we derive a single general bell inequality which is a sufficient and necessary condition for the correlation function for n particles to be describable in a local and realistic picture for the case in which measurements on each particle can be chosen between two arbitrary dichotomic observables we also derive a necessary and sufficient condition for an arbitrary n qubit mixed state to violate this inequality this condition is a generalization and reformulation of the horodecki family condition for two qubits

prediction of permeability for porous media reconstructed using multiple point statistics to predict multiphase flow through geologically realistic porous media it is necessary to have a three dimensional 3d representation of the pore space we use multiple point statistics based on two dimensional 2d thin sections as training images to generate geologically realistic 3d pore space representations thin section images can provide multiple point statistics which describe the statistical relation between multiple spatial locations and use the probability of occurrence of particular patterns assuming that the medium is isotropic a 3d image can be generated that preserves typical patterns of the void space seen in the thin sections the method is tested on berea sandstone for which a 3d image from micro ct computerized tomography scanning is available and shows that the use of multiple point statistics allows the long range connectivity of the structure to be preserved in contrast to two point statistics methods that tend to underestimate the connectivity furthermore a high resolution 2d thin section image of a carbonate reservoir rock is used to reconstruct 3d structures by the proposed method the permeabilities of the statistical images are computed using the lattice boltzmann method lbm the results are similar to the measured values to the permeability directly computed on the micro ct image for berea and to predictions using analysis of the 2d images and the effective medium approximation

positive and negative solutions of a quasi linear elliptic equation by a mountain pass method and truncature techniques abstract the existence of a positive and a negative solution of a problem of the type u f x u u 0 in u 0 on is proved when f has a growth at infinity depending both on u and on u the techniques are based on an iterative scheme of mountain pass approximated solutions and the use of a suitable truncature method

unified solver for rarefied and continuum flows with adaptive mesh and algorithm refinement this paper describes a unified flow solver ufs for rarefied and continuum gas flows the ufs separates the rarefied and continuum flow domains and selects appropriate solvers to combine the efficiency of continuum models with the accuracy of kinetic models the direct numerical solution of the boltzmann transport equation is used in rarefied regions while kinetic schemes of continuum fluid dynamics are used elsewhere using similar computational techniques for the kinetic and continuum solvers and employing intelligent domain decomposition algorithms attain the efficiency and numerical stability of the ufs solutions of test problems are presented to illustrate the capabilities of the ufs for high and low speed flows it is shown that the ufs can dynamically adapt the computational mesh and automatically introduce and remove kinetic patches to provide significant savings by limiting molecular scale solutions only to the regions where they are needed

positive solutions for nonlinear elliptic problems with dependence on the gradient abstract we consider a quasilinear neumann problem with a differential operator and a reaction term both dependent on u and du using topological methods together with suitable truncation and comparison techniques we show that the problem has at least one positive smooth solution

a semilinear elliptic pde not in divergence form via variational methods in this paper we consider a semilinear equation driven by an operator not in divergence form precisely the principal part of the operator is in divergence form but it has also a lower order term depending on du while the right hand side of the equation satisfies superlinear and subcritical growth conditions at zero and at infinity the problem has not a variational structure but despite that we use variational techniques in order to prove an existence and regularity result for the equation

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

locally refined discrete velocity grids for stationary rarefied flow simulations most of deterministic solvers for rarefied gas dynamics use discrete velocity or discrete ordinate approximations of the distribution function on a cartesian grid this grid must be sufficiently large and fine to describe the distribution functions at every space position in the computational domain for 3 dimensional hypersonic flows like in re entry problems this induces much too dense velocity grids that cannot be practically used for memory storage requirements in this article we present an approach to generate automatically a locally refined velocity grid adapted to a given simulation this grid contains much less points than a standard cartesian grid and allows us to make realistic 3 dimensional simulations at a reduced cost with a comparable accuracy

implicit unified gas kinetic scheme for steady state solutions in all flow regimes this paper presents an implicit unified gas kinetic scheme ugks for non equilibrium steady state flow computation the ugks is a direct modeling method for flow simulation in all regimes with the updates of both macroscopic flow variables and microscopic gas distribution function by solving the macroscopic equations implicitly a predicted equilibrium state can be obtained first through iterations with the newly predicted equilibrium state the evolution equation of the gas distribution function and the corresponding collision term can be discretized in a fully implicit way for fast convergence through iterations as well the lower upper symmetric gauss seidel lu sgs factorization method is implemented to solve both macroscopic and microscopic equations which improves the efficiency of the scheme since the ugks is a direct modeling method and its physical solution depends on the mesh resolution and the local time step a physical time step needs to be fixed before using an implicit iterative technique with a pseudo time marching step therefore the physical time step in the current implicit scheme is determined by the same way as that in the explicit ugks for capturing the physical solution in all flow regimes but the convergence to a steady state speeds up through the adoption of a numerical time step with large cfl number many numerical test cases in different flow regimes from low speed to hypersonic ones such as the couette flow cavity flow and the flow passing over a cylinder are computed to validate the current implicit method the overall efficiency of the implicit ugks can be improved by one or two orders of magnitude in comparison with the explicit one

an improved iterative hdg approach for partial differential equations abstract we propose and analyze an iterative high order hybridized discontinuous galerkin ihdg discretization for linear partial differential equations we improve our previous work 45 in several directions 1 the improved ihdg approach converges in a finite number of iterations for the scalar transport equation 2 it is unconditionally convergent for both the linearized shallow water system and the convection diffusion equation 3 it has improved stability and convergence rates 4 we uncover a relationship between the number of iterations and time stepsize solution order meshsize and the equation parameters this allows us to choose the time stepsize such that the number of iterations is approximately independent of the solution order and the meshsize and 5 we provide both strong and weak scalings of the improved ihdg approach up to 16 384 cores a connection between ihdg and time integration methods such as parareal and implicit explicit methods are discussed extensive numerical results for linear and nonlinear pdes are presented to verify the theoretical findings

conservative evaluation of boltzmann collision integral in discrete ordinates approximation abstract within the framework of a discrete ordinates approximation two conservative methods of evaluation of a boltzmann collision integral are proposed the first method that is partially conservative preserves density and kinetic momentum and is based on a decomposition of the velocity subspace on equal cubic cells centered around discrete velocity nodes the second method is completely conservative and needs for its construction the more elaborate projection technique described in the article both methods are tested on a numerical example and compared with other known methods it is demonstrated that a completely conservative method possesses better convergence properties with respect to numbers of discrete ordinates and integration nodes and is more efficient in a computational aspect

tight multipartite bell s inequalities involving many measurement settings which quantum states do not allow a local realistic lr description this question still remains open mainly because our present tools to test local realism are not optimal most of bell s inequalities are for the case in which only two measurement settings can be chosen by each observer e g the clauser horneshimony holt chsh inequality 1 inequalities for bipartite higher dimensional systems 2 multipartite bell s inequalities 3 6 one can call such inequalities standard ones one can expect that allowing the observers to choose between more than two observables should give more stringent constraints on lr models thus new inequalities may extend the class of nonseparable states which cannot be described by lr variables violation of local realism is an important ingredient for building quantum information protocols that decrease the communication complexity 7 and is a criterion for the efficient quantum key distribution 8 multisetting bell s inequalities may lead to such novel schemes although various nonstandard bell s inequalities are known 9 13 we still lack an efficient method for their derivation a general way is to define the facets of the correlation polytope 9 13 yet this is a computationally hard np problem 14 recently wu and zong 15 derived an inequality for three parties which involves four local settings for two observers and two settings for the third one and showed that it is stronger than the standard three particle bell inequalities here by generalizing the method of ref 15 we derive a wide class of tight bell s inequalities for n 2 parties and many measurement settings we find a single general bell s inequality that generates all inequalities from this class we also give a necessary and sufficient condition for the violation of the new inequalities by quantum predictions most importantly the inequalities reveal violation of local realism for the classes of states for which all standard bell s inequalities 5 6 are satisfied we start with the case ofn 3 observers suppose that the first two observers can choose between four settings and the third one between two settings we denote such problem as 4 4 2 l etai withi 2f 1 2 3 4g stand for the predetermined local realistic values for the first observer under the local settingi bj withj 2f 1 2 3 4g for similar values for the second observer and ck withk 2 f1 2g for the values for the third observer for the given run of the experiment we assume thata i b j a ndc k can take values 1 or 1 then the lr values satisfy the following algebraic identity a12 s 0 x k l 1 2 s 0 k l a1 1 k a 2 b1 1 l b 2 4

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 new shape preserving parallel thinning algorithm for 3d digital images this paper is concerned with a new parallel thinning algorithm for three dimensional digital images that preserves the topology and maintains their shape we introduce an approach of selecting shape points and outer layer used for erosion during each iteration the approach produces good skeleton for different types of corners the concept of using two image versions in thinning is introduced and its necessity in parallel thinning is justified the robustness of the algorithm under pseudo random noise as well as rotation with respect to shape properties is studied and the results are found to be satisfactory

efficiency of high order elements for continuous and discontinuous galerkin methods summary to evaluate the computational performance of high order elements a comparison based on operation count is proposed instead of runtime comparisons more specifically linear versus high order approximations are analyzed for implicit solver under a standard set of hypotheses for the mesh and the solution continuous and discontinuous galerkin methods are considered in two dimensional and three dimensional domains for simplices and parallelotopes moreover both element wise and global operations arising from different galerkin approaches are studied the operation count estimates show that for implicit solvers high order methods are more efficient than linear ones copyright 2013 john wiley sons ltd

application of discontinuous galerkin method to mechanical 2d problem with arbitrary polygonal and very high order finite elements abstract discontinuous galerkin with finite difference rules dgfd is applied to mechanical plane stress state problem the considered domain is discretized by polygonal mesh the polygonal elements can be for example a hexagon pentagon or just quadrangle or triangle they do not have to be convex and a fish mesh where the elements have fish shapes is used when the elements are rectangular then the orthogonality of chebyshev basis functions can be utilized in such a case very high order approximate solution can be obtained in this work the approximation order exceeds 10 and reaches 60 which in the latter case means 3600 numbers of degrees of freedom in a single element the paper is illustrated by a benchmark example in which the exact solution is recovered by dgfd method for various meshes in the other example the stress concentration is easily recovered by very high order version of dgfd method

efficient extraction of networks from three dimensional porous media 1 fluid flow through porous media and the thermal electrical and acoustic properties of these materials is largely controlled by the geometry and topology gt of the pore system which can be considered as a network network extraction techniques have been applied in many research fields including shape representation pattern recognition and artificial intelligence however the set of algorithms presented here significantly improves the efficiency of common thinning algorithms by introducing a sufficiency condition based on the idea of a simple set this paper describes an efficient and accurate algorithm for extracting the geometrical topological network that represents the pore structure of a porous medium referred to as the gt network the accurate medial axis and the specific gt description of the network are achieved by applying symmetrical and interval strategies during the erosion step in the image processing the gt network extraction algorithm presented here involves a number of steps including 1 calculation of the three dimensional euclidean distance map 2 clustering of voxels 3 extraction of the network of the pore space 4 partitioning of the pore space and 5 computation of shape factors the focus of this paper is mainly on the thinning method that underpins points 1 3 the paper is primarily a method description but we illustrate the functionality of the technique by extracting a pore scale gt network from micro computer tomography images of three sandstones

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