a gas kinetic scheme for the baer nunziato two phase flow model numerical methods for the baer nunziato bn two phase flow model have attracted much attention in recent years in this paper we present a new gas kinetic scheme for the bn two phase flow model containing non conservative terms in the framework of finite volume method in the view of microscopic aspect a generalized bhatnagar gross krook bgk model which matches with the bn model is constructed based on the integral solution of the generalized bgk model we construct the distribution functions at the cell interface then numerical fluxes can be obtained by taking moments of the distribution functions and non conservative terms are explicitly introduced into the construction of numerical fluxes in this method not only the complex iterative process of exact solutions is avoided but also the non conservative terms included in the equation can be handled well

an algorithmic approach to the construction of homomorphisms induced by maps in homology this paper is devoted to giving the theoretical background for an algorithm for computing homomorphisms induced by maps in homology the principal idea is to insert the graph of a given continuous map f into ag raph of a multi valued representable map f the multi valued representable maps have well developed continuity properties and admit a nite coding that per mits treating them by combinatorial methods we provide the construction of the homomorphism f induced by f such that f f the presented construction does not require subsequent barycentric subdivisions and simpli cial approximations of f the main motivation for this paper comes from the project of computing the conley index for discrete dynamical systems

study on gas kinetic unified algorithm for flows from rarefied transition to continuum the modified bgk equation adapted to various flow regimes can be presented by the aid of the basic characteristics on molecular movement and collision approaching to equilibrium the discrete velocity ordinate method is developed and applied to the velocity distribution function to remove its continuous dependency on the velocity space and then the velocity distribution function equation is cast into hyperbolic conservation law form with nonlinear source terms based on the unsteady time splitting method and the non oscillatory containing no free parameters and dissipative nnd scheme the gas kinetic finite difference second order scheme is constructed for the computation of the discrete velocity distribution functions the mathematical model on the interaction of molecules with solid surface is studied and used in the numerical method four types of numerical quadrature rules such as the modified gauss hermite formula the composite newton cotes integration method the gauss legendre numerical quadrature rule and the golden section number theoretic integral method are developed and applied to the discretized velocity space to evaluate the macroscopic flow parameters at each point in the physical space as a result a unified simplified gas kinetic algorithm is established for the flows from rarefied transition to continuum regime based on analyzing the inner parallel degree of the unified algorithm the parallel strategy adapted to the gas kinetic numerical algorithm is studied and then the hpf parallel processing software for the unified algorithm is developed to test the present method the one dimensional shock tube problems the flows past two dimensional circular cylinder and the flows around three dimensional sphere and spacecraft shape with various knudsen numbers are simulated the computational results are found in high resolution of the flow fields and good agreement with the theoretical dsmc n s and experimental results

homology computation by reduction of chain complexes abstract a new algorithm for computing the homology module of a finitely generated chain complex is given it is based on local one step reductions of the size of the initial chain complex and it has a clear geometrical interpretation the complexity of the algorithm is discussed in special cases

a multi speed compressible lattice boltzmann model an extension of the lattice boltzmann bgk method to compressible flows is presented that combines three novel additions 1 particles move density and energy weights in multiple velocity bins 11 for 1 d flow to nearby cell centers 2 the equilibrium distribution remains an unexpanded maxwellian and 3 transport and relaxation to equilibrium are performed implicitly at each node these advances allow for the parallel modeling of high mach number shocks and high reynolds number flows while avoiding advective numerical diffusion the need for riemann solvers and non linear limiters a 1d shock tube application is shown generalization to higher dimensions and multi materials are discussed

gas kinetic schemes for compressible flow of real gases abstract this paper extends gas kinetic schemes to the euler equations for real gases in the current scheme the maxwell boltzmann gas distribution function is modified to recover macroscopic flow equations more specifically the internal degree of freedom of the gas distribution function becomes a function of flow variables according to the general equation of state the numerical results confirm the accuracy and robustness of the gas kinetic approach

gas kinetic numerical study of complex flow problems covering various flow regimes the boltzmann simplified velocity distribution function equation as adapted to various flow regimes is described on the basis of the boltzmann shakhov model from the kinetic theory of gases in this study the discrete velocity ordinate method of gas kinetic theory is studied and applied to simulate complex multi scale flows on the basis of using the uncoupling technique on molecular movements and collisions in the dsmc method the gas kinetic finite difference scheme is constructed by extending and applying the unsteady time splitting method from computational fluid dynamics which directly solves the discrete velocity distribution functions the gauss type discrete velocity numerical quadrature technique for flows with different mach numbers is developed to evaluate the macroscopic flow parameters in the physical space as a result the gas kinetic numerical algorithm is established for studying the three dimensional complex flows with high mach numbers from rarefied transition to continuum regimes on the basis of the parallel characteristics of the respective independent discrete velocity points in the discretized velocity space a parallel strategy suitable for the gas kinetic numerical method is investigated and then the hpf high performance fortran parallel programming software is developed for simulating gas dynamical problems covering the full spectrum of flow regimes to illustrate the feasibility of the present gas kinetic numerical method and simulate gas transport phenomena covering various flow regimes the gas flows around three dimensional spheres and spacecraft like shapes with different knudsen numbers and mach numbers are investigated to validate the accuracy of the numerical methods through hpf parallel computing the computational results determine the flow fields in high resolution and agree well with the theoretical and experimental data this computing in practice has confirmed that the present gas kinetic algorithm probably provides a promising approach for resolving hypersonic aerothermodynamic problems with the complete spectrum of flow regimes from the gas kinetic point of view for solving the mesoscopic boltzmann model equation

kinetic flux vector splitting for the navier stokes equations before a hybrid scheme can be developed combining the direct simulation monte carlo dsmc method and a navier stokes ns representation one must have access to compatible kinetic split fluxes from the ns portion of the hybrid scheme the kinetic theory basis is given for the development of the required fluxes from the chapman enskog velocity distribution function for a simple gas and these are then extended to a polyatomic gas by use of the eucken approximation the derived fluxes are then used to implement boundary conditions at solid surfaces that are based on concepts associated with kinetic theory and the dsmc method this approach is shown to lead to temperature slip and velocity slip as a natural outcome of the new formulation a requirement for use in the near continuum regime where dsmc and ns must be joined several different flows for which solid boundaries are not present are computed using the derived fluxes together with a second order finite volume scheme and the results are shown to agree well with several established numerical schemes for the ns equations

the fractal property of the lorenz attractor in a 1963 paper lorenz inferred that the lorenz attractor must be an infinite complex of surfaces we investigate this fractal property of the lorenz attractor in two ways firstly we obtain explicit plots of the fractal structure of the lorenz attractor using symbolic dynamics and multiple precision computations of periodic orbits the method we derive for multiple precision computation is based on iterative refinement and can compute even highly unstable periodic orbits with long symbol sequences with as many as 100 digits of accuracy ordinary numerical integrations are much too crude to show even the coarsest splitting of surfaces and there appear to be no other explicit computations of the fractal structure in the extensive literature about the lorenz attractor secondly we apply a well known formula that gives the hausdorff dimension of the lorenz attractor in terms of the characteristic multipliers of its unstable periodic orbits the formula converges impressively and the hausdorff dimension of the lorenz attractor appears to be 2 0627160 we use comparison with explicit computations of the fractal structure and discuss the accuracy of this formula and its applicability to the lorenz equations additionally we apply periodic orbit theory to the lorenz attractor and exhibit its spectral determinant and compute its lyapunov exponent

example of a physical system with a hyperbolic attractor of the smale williams type a simple and transparent example of a nonautonomous flow system with a hyperbolic strange attractor is suggested the system is constructed on the basis of two coupled van der pol oscillators the characteristic frequencies differ twice and the parameters controlling generation in both oscillators undergo a slow periodic counterphase variation in time in terms of stroboscopic poincare sections the respective 4d mapping has a hyperbolic strange attractor of the smale williams type qualitative reasoning and quantitative data of numerical computations are presented and discussed e g lyapunov exponents and their parameter dependencies a special test for hyperbolicity based on analysis of distributions of angles between stable and unstable subspaces of a chaotic trajectory is performed

green function estimate for censored stable processes sharp two sided estimates for green functions of censored stable process y in a bounded c 1 1 open set d are obtained where 1 2 it is shown that the martin boundary and minimal martin boundary of y can all be identified with the euclidean boundary of d sharp two sided estimates for the martin kernel of y are also derived

numerical hydrodynamics from gas kinetic theory a new high resolution numerical hydrodynamic scheme is developed from considerations of gas kinetic theory this boltzmann type scheme uses the particle distribution function and follows its evolution to evaluate the numerical fluxes the intrinsic multidimensional property in the evolution stage of gas distribution function distinguishes it from any other high order upwind scheme from approximate boltzmann equation the navier stokes equations can be derived and this new numerical method uses this advantage to approximate the solutions of the navier stokes equations where the advection and dissipation effects are coupled in the time dependent gas distribution function several well defined numerical test cases are presented in one and two dimensions and it is shown that the resolution in the numerical results is competitive with those of any of the current high resolution schemes without special treatment of the discontinuities for strong rarefaction waves our scheme gives better results in this thesis we try to separate clearly numerical and physical effects and we also give a concise explanation of the necessity of including numerical viscosities implicitly or explicitly in high resolution godunov type and central difference schemes even for the navier stokes equations the main part of this thesis will be published in two papers 27 and 45 in the journal of computational physics

positive periodic solutions of second order nonlinear equations with indefinite weight multiplicity results and complex dynamics we prove the existence of a pair of positive t periodic solutions as well as the existence of positive subharmonic solutions of any order and the presence of chaotic like dynamics for the scalar second order ode u a t g u 0 where g x is a positive function on r superlinear at zero and sublinear at infinity and a t is a t periodic and sign indefinite weight of the form a t a t with 0 and large

regular article a new discrete velocity method for navier stokes equations the relation between the lattice boltzmann method which has recently become popular and the kinetic schemes which are routinely used in computational fluid dynamics is explored a new discrete velocity method for the numerical solution of navier stokes equations for incompressible fluid flow is presented by combining both the approaches the new scheme can be interpreted as a pseudo compressibility method and for a particular choice of parameters this interpretation carries over to the lattice boltzmann method

gas kinetic scheme with discontinuous derivative for low speed flow computation the present paper concerns the improvement of the gas kinetic scheme gks for low speed flow computation in the modified gks scheme the flow distributions with discontinuous derivatives are used as an initial condition at the cell interface for the flux evaluation this discontinuity is determined by considering both the flow characteristic and grid s resolution compared with gks method with a continuous slope for the flow variables at a cell interface the new scheme is more robust and accurate in the under resolved flow computation the new scheme presents much less numerical oscillation the extension of the current scheme to unstructured mesh is straightforward to validate the method both computations of 2d lid driven cavity flow and 3d flow past a sphere are performed the numerical results validate the current method

chaos in the lorenz equators a computer assisted proof part ii details details of a new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems is described the technique combines abstract existence results based on the conley index theory with rigorous computer assisted computations as an application of these methods it is proven that for some explicit parameter values the lorenz equations exhibit chaotic dynamics

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

gas kinetic numerical studies of three dimensional complex flows on spacecraft re entry the gas kinetic numerical algorithm solving the boltzmann model equation is extended and developed to study the three dimensional hypersonic flows of spacecraft re entry into the atmosphere in perfect gas in this study the simplified velocity distribution function equation for various flow regimes is presented on the basis of the kinetic boltzmann shakhov model the discrete velocity ordinate technique and numerical quadrature methods such as the gauss quadrature formulas with the weight function 2 p 1 2exp v 2 and the gauss legendre numerical quadrature rule are studied to resolve the barrier in simulating complex flows from low mach numbers to hypersonic problems specially the gas kinetic finite difference scheme is constructed for the computation of three dimensional flow problems which directly captures the time evolution of the molecular velocity distribution function the gas kinetic boundary conditions and numerical procedures are studied and implemented by directly acting on the velocity distribution function the hpf high performance fortran parallel implementation technique for the gas kinetic numerical method is developed and applied to study the hypersonic flows around three dimensional complex bodies the main purpose of the current research is to provide a way to extend the gas kinetic numerical algorithm to the flow computation of three dimensional complex hypersonic problems with high mach numbers to verify the current method and simulate gas transport phenomena covering various flow regimes the three dimensional hypersonic flows around sphere and spacecraft shape with different knudsen numbers and mach numbers are studied by hpf parallel computing excellent results have been obtained for all examples computed

leray functor and cohomological conley index for discrete dynamical systems we introduce the leray functor on the category of graded modules equipped with an endomorphism of degree zero and we use this functor to define the cohomological conley index of an isolated invariant set of a homeomorphism on a locally compact metric space we prove the homotopy and additivity properties for this index and compute the index in some examples as one of applications we prove the existence of nonconstant bounded solutions of the euler approximation of a certain system of ordinary differential equations

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

a robust and accurate led bgk solver on unstructured adaptive meshes starting from the bgk model of the boltzmann equation we develop a robust and accurate finite volume gas kinetic scheme on unstructured triangular meshes the proposed numerical approach is composed of two steps an initial reconstruction step and a gas evolution step in the initial reconstruction step an unstructured version of the local extremum diminishing interpolation is applied to the conservative variables and to compute left and right states along a node edge in the gas evolution step the local integral solution of the bgk model is used to compute numerical fluxes at a cell interface this approach provides an alternative to riemann solvers and yields numerical schemes which possess many desirable properties that may not be found in godunov type schemes a classich refinement adaptive procedure is implemented to increase the spatial resolution of high speed unsteady flow characteristics such as shock waves contact discontinuities or expansion waves with minimal computational costs and memory overheads this procedure involves mesh enrichment coarsening steps to either insert nodes on an edge center in high gradient regions or delete nodes in over resolved regions numerical results of several test cases for unsteady compressible inviscid flows are presented in order to verify the accuracy and robustness of the current numerical approach the computed results are compared with analytical solutions experimental data the results of structured mesh calculations and the results obtained by widely used flux splitting methods

low speed flow simulation by the gas kinetic scheme this paper extends the gas kinetic bgk type scheme to low mach number flows and thus shows that incompressible flow solutions are accurately obtained from the bgk scheme in the low mach number limit the influence of boundary conditions internal molecular degrees of freedomk and the flow mach numbermon the accuracy of the solutions of incompressible or nearly incompressible flow problems is quantitatively evaluated the gas kinetic scheme is tested carefully in two numerical examples namely the cavity flow problem and the flow passing a backward facing step problem for the cavity flow problem the numerical results from the gas kinetic scheme under different reynolds numbers compare well with ghia s data for the backward step problem the numerical results are compared accurately with previously published experimental data

rkdg finite element method combined with bgk scheme for solving fluid dynamics system this paper presents a new type of second order scheme for solving the system of euler equations which combines the runge kutta discontinuous galerkin rkdg finite element method and the bhatnagar gross krook bgk scheme we first discretize the euler equations in space with the dg method and then the resulting system will be discretized using an rk method finally a second order bgk method is used to construct the numerical flux the proposed scheme keeps the main advantages of the dg finite element method including its flexibility in handling irregular solution domains and in parallelization moreover with the bgk approach the proposed schemes avoid using riemann solvers the efficiency and effectiveness of the proposed method are illustrated by several numerical examples in one and two dimensional spaces

detection of renewal system factors via the conley index let n be an isolating neighborhood for a map f if we can decompose n into the disjoint union of compact sets n 1 and n 2 then we can relate the dynamics on the maximal invariant set inv n to the shift on two symbols by noting which component of n each iterate of a point x inv n lies in we examine a method based on work by mischaikow szymczak et al for using the discrete conley index to detect explicit subshifts of the shift associated to n in essence we measure the difference between the conley index of inv n and the sum of the indices of inv n 1 and inv n 2

a conley index study of the evolution of the lorenz strange set abstract in this paper we study the lorenz equations using the perspective of the conley index theory more specifically we examine the evolution of the strange set that these equations posses throughout the different values of the parameter we also analyze some natural morse decompositions of the global attractor of the system and the role of the strange set in these decompositions we calculate the corresponding morse equations and study their change along the successive bifurcations we see how the main features of the evolution of the lorenz system are explained by properties of the dynamics of the global attractor in addition we formulate and prove some theorems which are applicable in more general situations these theorems refer to poincare andronov hopf bifurcations of arbitrary codimension bifurcations with two homoclinic loops and a study of the role of the traveling repellers in the transformation of repeller attractor pairs into attractor repeller ones

new developments in kinetic schemes kinetic schemes for the numerical solution of pde of fluid dynamics are a new class of upwind schemes the moment method strategy used in constructing these schemes is based on the connection between the boltzmann equation of kinetic theory of gases and governing equations of fluid dynamics any upwind scheme for the solution of the boltzmann equation becomes an upwind scheme for the solution of pde of fluid dynamics by taking suitable moments this idea has been exploited in the present paper in developing least squares kinetic upwind method lskum and kinetic flux vector splitting method on moving grid kfmg the robustness and versatility of lskum has been demonstrated by applying it to 2 d flow problems of inviscid gas dynamics the lskum has been found to operate on any type of grid and can be called a grid fault tolerant scheme in view of the use of least squares approximation to the space derivatives the kfmg is a promising new method showing how easily the kinetic schemes lend themselves easily to problems involving moving grids which are generally employed while solving problems in unsteady aerodynamics