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

melnikov method for discontinuous planar systems abstract this paper treats the occurrence of homoclinic solutions in planar systems with discontinuous right hand side more precisely we deal with a t periodic perturbed system such that the unperturbed system is an autonomous possessing homoclinic orbit by means of the so called non smooth melnikov function there is shown the existence of a homoclinic solution for a perturbed system the non smooth melnikov function is derived and the method of how to find it in concrete problems is also introduced

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

periodic occurrence of chaotic behavior of homoclinic tangles abstract in this article we illustrate through numerical simulations some important aspects of the dynamics of the periodically perturbed homoclinic solutions for a dissipative saddle more explicitly we demonstrate that when homoclinic tangles are created three different dynamical phenomena namely horseshoes periodic sinks and attractors with sinai ruelle bowen measures manifest themselves periodically with respect to the magnitude of the forcing function in addition when the stable and the unstable manifolds are pulled apart so as not to intersect first rank 1 attractors then quasi periodic attractors are added to the dynamical scene

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

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

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

bifurcations of subharmonics abstract we consider a piecewise linear second order ordinary differential equation which is topologically equivalent to the sine gordon equation the system is subjected to a time harmonic disturbance and the behavior of the periodic solutions is examined it is shown that subharmonic motions with period n times that of the disturbance appear via saddle node bifurcations for all n 1 2 3 these motions then undergo period doubling or pitchfork bifurcations as parameters are varied beyond the saddle node bifurcation values the limit n is considered and is compared with a melnikov calculation for the system due to the piecewise linear nature of the system no approximations are necessary and thus the results are valid for nonsmall parameter values furthermore this provides an answer to the uniformity question in the study of stability of subharmonics

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

melnikov method for homoclinic bifurcation in nonlinear impact oscillators based on an inverted pendulum impacting on rigid walls under external periodic excitation a class of nonlinear impact oscillators is discussed for its homoclinic bifurcation the melnikov method established for smooth dynamical systems is extended to be applicable to the nonsmooth one for nonlinear impact systems closed form solutions between impacts are generally unavailable the absence of closed form solutions makes difficulties in estimation of the gap between the stable manifold and unstable manifold in this paper we give a method to compute the melnikov functions up to the n t h order so as to obtain conditions of parameters for the persistence of homoclinic cycles which are formed via the identification given by the impact rule

density dependent limits for a nonlinear reaction diffusion model a reaction diffusion model constructed on a grid by linking nonlinear density dependent birth and death processes through particle diffusion is studied by proving two laws of large numbers and a fluctuation theorem under varying assumptions on the density parameter

normal form maps for grazing bifurcations in n dimensional piecewise smooth dynamical systems abstract this paper presents a unified framework for performing local analysis of grazing bifurcations in n dimensional piecewise smooth systems of odes these occur when a periodic orbit has a point of tangency with a smooth n 1 dimensional boundary dividing distinct regions in phase space where the vector field is smooth it is shown under quite general circumstances that this leads to a normal form map that contains to lowest order either a square root or a 3 2 type singularity according to whether the vector field is discontinuous or not at the grazing point in particular contrary to what has been reported in the literature piecewise linear local maps do not occur generically first the concept of a grazing bifurcation is carefully defined using appropriate non degeneracy conditions next complete expressions are derived for calculating the leading order term in the normal form poincare map at a grazing bifurcation point in arbitrary systems using the concept of a discontinuity mapping finally the theory is compared with numerical examples including bilinear oscillators a relay feedback controller and general third order systems

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

bifurcation and chaos near sliding homoclinics we study the chaotic behaviour of a time dependent perturbation of a discontinuous differential equation whose unperturbed part has a sliding homoclinic orbit that is a solution homoclinic to a hyperbolic fixed point with a part belonging to a discontinuity surface we assume the time dependent perturbation satisfies a kind of recurrence condition which is satisfied by almost periodic perturbations following a functional analytic approach we construct a melnikov like function m in such a way that if m has a simple zero at some point then the system has solutions that behave chaotically applications of this result to quasi periodic systems are also given

impact oscillators with homoclinic orbit tangent to the wall abstract homoclinic bifurcation for a nonlinear inverted pendulum impacting between two rigid walls under external periodic excitation is analyzed under the hypothesis that the unperturbed system has a homoclinc orbit tangent to the wall consequently the impact surface cannot be chosen as the poincare section to measure the distance between the perturbed stable and unstable manifolds furthermore compared to the case that the unperturbed homoclinic orbit intersects the wall transversally more cases are involved as the parameters vary thus the analysis of the homoclinic orbit tangent to the wall is much more difficult in this paper by using a method of melnikov type we derive sufficient conditions under which the perturbed stable and unstable manifolds intersect transversally as an application an impact oscillator of duffing type is studied in detail

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

bifurcations in piecewise smooth feedback systems this paper is concerned with the analysis and classification of non standard bifurcations in piecewise smooth feedback systems bifurcations involving fixed points in maps equilibria and limit cycles in flows are considered with particular attention to border collisions of fixed points in maps non smooth bifurcations of equilibria grazing bifurcations and sliding bifurcations of limit cycles in flows the aim is to describe existing and novel results to form the basis of a consistent theory of bifurcations in such systems in so doing a novel approach to classify non smooth bifurcations of equilibria in flows is presented experimental results on pulse width modulated voltage mode controlled dc dc power converters motivates the relevance to applications of the analysis proposed

on the origin and bifurcations of stick slip oscillations a recently proposed model of macroscopic friction is investigated using methods of dynamical systems analysis particular emphasis is put on the bifurcations associated with the appearance of stick

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

homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations nonautomonous ordinary differential equations depending on two parameters 1 and 2 are considered in rn it is assumed that when both parameters are zero the differential equation is autonomous with a hyperbolic equilibrium and a homoclinic solution no restriction is placed on the dimension of the phase space rn or on the dimension of intersection of the stable and unstable manifolds by means of the method of lyapunov schmidt a bifurcation function h is constructed between two finite dimensional spaces where the zeros of h correspond to homoclinic solutions at nonzero parameter values the independent variables of h consist of scalars 1 2 and a vector where is a phase angle and corresponds to directions other than along the original homoclinic solution tangent to both the stable and unstable manifolds when is fixed the equation h 0 yields in general several bifurcation curves through the origin in the 1 2 plane along which there exists a homoclinic solution when is varied these become a number of wedge shaped regions the theory is applied to two examples one in r6 where the invariant manifolds meet in dimension three and a second in r4 where these manifolds agree

nonsmooth homoclinic orbits melnikov functions and chaos in discontinuous systems abstract we first study the problem of chaotic behaviour in time perturbed discontinuous systems whose unperturbed part has a piecewise c 1 homoclinic solution transversally crossing the discontinuity manifolds we show that if a certain melnikov function has a simple zero at some point then the system has solutions that behave chaotically the melnikov function is explicitly constructed for two dimensional systems and extends the usual melnikov function for the smooth case in the second part we extend these results to sliding homoclinic bifurcations we also mention some possibilities for further research

on hopf bifurcation in non smooth planar systems abstract as we know for non smooth planar systems there are foci of three different types called focus focus ff focus parabolic fp and parabolic parabolic pp type respectively the poincare map with its analytical property and the problem of hopf bifurcation have been studied in coll et al 2001 3 and filippov 1988 6 for general systems and in zou et al 2006 13 for piecewise linear systems in this paper we also study the problem of hopf bifurcation for non smooth planar systems obtaining new results more precisely we prove that one or two limit cycles can be produced from an elementary focus of the least order order 1 for foci of ff or fp type and order 2 for foci of pp type theorem 2 3 different from the case of smooth systems for piecewise linear systems we prove that 2 limit cycles can appear near a focus of either ff fp or pp type theorem 3 3

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

on the derivation of reaction diffusion equations as limit dynamics of systems of moderately interacting stochastic processes we consider systems of moderately interacting particles which are divided into a finite number of different subpopulations and show that in the limit as the population size tends to infinity the empirical processes of the subpopulations converge to the solution of a system of reaction diffusion equations

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

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

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

a stochastic model of a chemical reaction with diffusion stochastic systems of brownian motions with multiple deletion of particles are introduced to model a chemical reaction with diffusion convergence to the solution of a deterministic nonlinear reaction diffusion equation is proved without high density assumptions

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

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

continuous and discontinuous grazing bifurcations in impacting oscillators this paper seeks to formulate conditions for the persistence of a local attractor in the immediate vicinity of periodic and quasiperiodic grazing trajectories in an impacting mechanical system a local analysis based on the discontinuity mapping approach is employed to derive a normal form description of the dynamics near the grazing trajectory in agreement with previous studies of grazing periodic trajectories it is found that the catastrophic loss of a local attractor and strong instability characteristic of grazing bifurcations is directly associated with the repeated application of a square root term that appears to lowest order in the normal form expansion specifically it is found that the square root term is absent in the description of the dynamics normal to a quasiperiodic trajectory covering a co dimension one invariant torus resulting in a piecewise linear description of the normal dynamics and at most a weak instability in contrast for co dimension two or higher the square root term is generically present in the normal dynamics here however the quasiperiodicity of the grazing motion implies that there is no upper limit to the time between impacts on nearby trajectories suggesting the persistence of a local attractor for some interval about the parameter value corresponding to grazing the results of the analysis are illustrated through a series of model examples

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

exponential dichotomies heteroclinic orbits and melnikov functions the interest in the behaviour of the solutions to perturbed autonomous systems with bounded perturbative term like