discretization on non orthogonal quadrilateral grids for inhomogeneous anisotropic media two classes of discretization methods are proposed for control volume formulations on quadrilateral grids in two space dimensions curvilinear grids are considered as a special case the methods are applicable for any system of conservation laws where the flux is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy and or non orthogonality of the grid further properties of the methods will be discussed and examples of their use will be presented

brief paper design of highly synchronizable and robust networks in this paper the design of highly synchronizable sparse and robust dynamical networks is addressed better synchronizability means faster synchronization of the oscillators sparsity means a low ratio of links per nodes and robustness refers to the resilience of a network to the random failures or intentional removal of some of the nodes links golden spectral dynamical networks graphs are those for which the spectral spread the difference between the largest and smallest eigenvalues of the adjacency matrix is equal to the spectral gap the difference between the two largest eigenvalues of the adjacency matrix multiplied by the square of the golden ratio these networks display the property of small worldness are very homogeneous and have large isoperimetric expansion constant together with a very high synchronizability and robustness to failures of individual oscillators in particular the regular bipartite dynamical networks reported here by the first time have the best possible expansion and consequently are the most robust ones against node link failures or intentional attacks

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

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

synchronizable nodes in networked systems synchronization appears in networked systems as a phenomenon of interacting nodes the notion of synchronizability describes the property that the nodes can be forced to a synchronous behaviour by an appropriate interconnection structure whereas in earlier papers synchronizability has been investigated in terms of both the properties of the nodes and of the interconnection topology this paper shows that for linear coupled systems synchronizability is a property of the node dynamics alone if the nodes do not satisfy the necessary and sufficient synchronizability condition derived no interaction structure can ensure a synchronous behaviour of the networked system on the other hand if this condition is satisfied several interconnection structures given in this paper ensure synchronism globally in the whole state space

multigrid solution of the euler equations using implicit schemes on utilise une methode multigrille du type factorisation approchee couplee a un schema de direction alternee implicite pour resoudre les equations d euler pour l ecoulement transsonique sur un profil aerodynamique on obtient une convergence tres rapide

a multipoint flux mixed finite element method we develop a mixed finite element method for single phase flow in porous media that reduces to cell centered finite differences on quadrilateral and simplicial grids and performs well for discontinuous full tensor coefficients motivated by the multipoint flux approximation method where subedge fluxes are introduced we consider the lowest order brezzi douglas marini bdm mixed finite element method a special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell centered system for the pressures theoretical and numerical results indicate second order convergence for pressures at the cell centers and first order convergence for subedge fluxes second order convergence for edge fluxes is also observed computationally if the grids are sufficiently regular

first order convergence of multi point flux approximation on triangular grids and comparison with mixed finite element methods in this paper we show first order convergence of a multi point flux approximation control volume method mpfa on unstructured triangular grids in this approach the flux approximation is derived directly in the physical space in order to do this we introduce a perturbed mixed finite element method that is equivalent to the mpfa scheme and prove the first order convergence of this approach moreover we carefully compare the computational performance properties of the mpfa method with those of a lowest order raviart thomas and brezzi douglas marini mixed finite element approximation

switching controllability of discrete time multi agent systems with multiple leaders and time delays this paper investigates the switching controllability of discrete time multi agent systems with multiple leaders and time delays on undirected networks in which both a single time delay and multiple time delays are addressed it is established that the switching controllability of discrete time multi agent systems is only determined by the information from the leaders to the followers and the discrete time multi agent systems are controllable even if each of its subsystems is uncontrollable for the special case of fixed topology an equivalent augmented system without time delays is proposed to analyze the controllability of multi agent systems via pbh rank test some numerical examples are provided to demonstrate the effectiveness of the theoretical results

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

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

modified kinetic flux vector splitting schemes for compressible flows we investigate the traditional kinetic flux vector splitting kfvs and bgk schemes for the compressible euler equations first based on a careful study of the behavior of the discrete physical variables across the contact discontinuity we analyze quantitatively the mechanism of inducing spurious oscillations of the velocity and pressure in the vicinity of the contact discontinuity for the first order kfvs and bgk schemes then with the help of this analysis we propose a first order modified kfvs mkfvs scheme which is oscillation free in the vicinity of the contact discontinuity provided certain consistent conditions are satisfied moreover by using piecewise linear reconstruction and van leer s limiter the first order mkfvs scheme is extended to a second order one consequently a nonoscillatory second order mkfvs scheme is constructed finally by combing the mkfvs schemes with the model we successfully extend the mkfvs schemes to multi flows and propose therefore a first and second order mkfvs schemes for multi fluid computations which are nonoscillatory across fluid interfaces a number of numerical examples presented in this paper validate the theoretic analysis and demonstrate the good performance of the mkfvs schemes in simulation of contact discontinuities for both single and multi fluids

synchronization of harmonic oscillators under restorative coupling with applications in electrical networks the role of restorative coupling on synchronization of coupled identical harmonic oscillators is studied necessary and sufficient conditions under which the individual systems solutions converge to a common trajectory are presented through simple physical examples the meaning and limitations of the theorems are expounded also to demonstrate their versatility the results are extended to cover lti passive electrical networks one of the extensions generalizes the well known link between the asymptotic stability of the synchronization subspace and the second smallest eigenvalue of the laplacian matrix

brief paper synchronization of coupled harmonic oscillators in a dynamic proximity network in this paper we revisit the synchronization problems for coupled harmonic oscillators in a dynamic proximity network unlike many existing algorithms for distributed control of complex dynamical networks that require explicit assumptions on the network connectivity we show that the coupled harmonic oscillators can always be synchronized without imposing any network connectivity assumption moreover we also investigate the synchronization with a leader and show that all harmonic oscillators can asymptotically attain the position and velocity of the leader again without any assumption on connectivity of the followers numerical simulation illustrates the theoretical results

a multidimensional gas kinetic bgk scheme for hypersonic viscous flow this paper concerns the development and application of a gas kinetic scheme based on the bhatnagar gross krook bgk model for the navier stokes equations in the study of hypersonic viscous flow firstly we extend the gas kinetic navier stokes solver k xu a gas kinetic bgk scheme for the navier stokes equations and its connection with artificial dissipation and godunov method j comput phys 171 2001 289 335 by implementing a multidimensional particle propagation mechanism in the flux evaluation where the gradients of flow variables in both normal and tangential directions of a cell interface are explicitly included with the construction of a time averaged flux function an implicit bgk scheme with lu sgs method is constructed the main purpose of the current research is to pave the way to extend the current approach directly to the flow computation with unstructured mesh where the flow gradients in both parallel and perpendicular directions around a cell interface can be explicitly taken into account in a viscous flow computation in the numerical parts we concentrate on the computation of heat flux in laminar hypersonic viscous flows where complicated flow phenomena i e shock boundary layer interaction flow separation and viscous inviscid interaction will be encountered the cases studied include the type iv shock shock interaction around a circular cylinder and hypersonic flow passing through a double cone geometry in the hypersonic viscous flow in comparison with the capturing of velocity and pressure fields the accurate computation of stress and heat flux bears large difficulties in all cases studied here the heat fluxes obtained across body surfaces have good agreement with the experimental measurements

compact quasi gas dynamic system and high performance computing abstract new compact hyperbolic quasi gas dynamic system of equations chqgd is derived from kinetic models for the one particle distribution function the system differs from the navier stokes equations by a second order value in the knudsen number the presence of second time derivatives makes the chqgd system hyperbolic in contrast to the navier stokes equations the system of chqgd equations allows one to construct efficient including parallel computational algorithms for modeling viscous flows we consider the properties of the new compact hyperbolic chqgd system and methods for its numerical solution two discretization options are described based on the finite volume method estimates are given of the solution dependence on the small parameter at the second time derivative special attention is paid to the algorithms adaptation to the architecture of computing systems with massive parallelism as an example of the algorithms use the results of numerical simulation of a supersonic separated flow around a compression angle are given a comparison with experimental data and numerical solutions of the navier stokes equations is presented

aerodynamic shape optimization using computational fluid dynamics and parallel simulated annealing algorithms aerodynamic shape design using stochastic optimization methods such as simulated annealing method to optimize objective functions evaluated by modern state of the art computational fluid dynamics solvers normally requires enormous computation time to search for the global optimal design aerodynamic shape optimization of internal flow systems is studied using euler navier stokes solvers and parallel simulated annealing algorithm which is implemented on parallel computing platforms a variety of inverse and direct design of internal flow systems are carried out to examine the efficiency and speedup of the parallel simulated annealing algorithms

module phase synchronization in complex dynamic system the concept of module phase synchronization was proposed the chaos synchronization between drive system and response system was achieved in module space and phase space respectively module phase synchronization different from the evolutions in real space there is no pseudorandom behavior in phase space when module phase synchronization achieve all the phases of complex state variables switched between two fixed values which are determined by initial values of drive system and the modules varied within a bounded field the theoretical analysis and the simulations were also given

second order global consensus in multiagent networks with random directional link failure in this paper we consider the second order globally nonlinear consensus in a multiagent network with general directed topology and random interconnection failure by characterizing the behavior of stochastic dynamical system with the corresponding time averaged system a criterion for the second order consensus is derived by constructing a lyapunov function for the time averaged network by associating the solution of random switching nonlinear system with the constructed lyapunov function a sufficient condition for second order globally nonlinear consensus in a multiagent network with random directed interconnections is also established it is required that the second order consensus can be achieved in the time averaged network and the lyapunov function decreases along the solution of the random switching nonlinear system at an infinite subsequence of the switching moments a numerical example is presented to justify the correctness of the theoretical results

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

comparison of implicit schemes for the incompressible navier stokes equations for a computational flow simulation tool to be useful in a design environment it must be very robust and efficient to develop such a tool for incompressible flow applications a number of different implicit schemes are compared for several two dimensional flow problems in the current study the schemes include point jacobi relaxation gauss seidel line relaxation incomplete lower upper decomposition and the generalized minimum residual method preconditioned with each of the three other schemes the efficiency of the schemes is measured in terms of the computing time required to obtain a steady state solution for the laminar flow over a backward facing step the flow over a naca 4412 airfoil and the flow over a three element airfoil using overset grids the flow solver used in the study is the ins2d code that solves the incompressible navier stokes equations using the method of artificial compressibility and upwind differencing of the convective terms the results show that the generalized minimum residual method preconditioned with the incomplete lower upper factorization outperforms all other methods by at least a factor of 2

algebraic criteria for second order global consensus in multi agent networks with intrinsic nonlinear dynamics and directed topologies this paper discusses the second order globally nonlinear consensus in general multi agent directed networks with both non time delay and time delay couplings some easily manageable delay independent algebraic criteria are presented to unravel the underlying mechanics for reaching the second order consensus the obtained results are associated with the underlying network interactive topologies inner coupling matrices and coupling gains between agents theoretical derivation is complemented by its validation on a simulation example

m matrix flux splitting for general full tensor discretization operators on structured and unstructured grids diagonal tensor flux approximations are commonly used in fluid dynamics this approximation introduces an o 1 error in flux whenever the coordinate system is nonaligned with the principal axes of the tensor which is particularly common when employing curvilinear gridding in general a consistent full tensor flux approximation leads to a significant increase in support and consequent size of the jacobian matrix after decomposition of a general full tensor flux into a diagonal tensor flux together with cross terms time split semi implicit stable full tensor flux approximations are introduced with in a general finite volume formalism enabling the standard diagonal tensor jacobian matrix structure to be retained for single phase flow impes and standard block fully implicit formulations while ensuring spatial consistency of the discretization for both structured and unstructured grids stability of the scheme is proven for constant elliptic coefficients the results presented demonstrate the benefits of the method for multiphase flow within a fully implicit framework on structured and unstructured grids

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

synchronization of coupled discrete time harmonic oscillators with rational frequency this technical note studies the synchronization of coupled discrete time harmonic oscillators with rational frequency under switching topology the remarkable feature of this problem is that the conditions that merely involve the connectivity structure of topology does not suffice for synchronizing the oscillators we propose a frequency dependent topology condition that indicates in what way the topology switches and introduce firmly nonexpansive mapping fnm from functional analysis under the condition the synchronization of coupled oscillators is related to an infinite product of fnms which share only one zero common fixed point by a convergence result on infinite product of a finite number of fnm we present a synchronization result for the coupled oscillators finally a simulation example is given to illustrate the effectiveness of the result

containment control for coupled harmonic oscillators with multiple leaders under directed topology this paper investigates the problem of containment control for coupled harmonic oscillators with multiple leaders under directed topology using tools from matrix graph and stability theories necessary and sufficient conditions are obtained for coupled harmonic oscillators under continuous time and sampled data based protocols respectively when the continuous time protocol is used it is proved that every follower will ultimately converge to the convex hull spanned by the leaders if and only if there exists at least one leader that has a directed path to that follower at any time when the sampled data based protocol is used it is shown that the containment can be achieved if and only if 1 an appropriate sampling period is chosen and 2 for every follower there exists at least one leader that has a directed path to that follower at any time and we also give the containment conditions for coupled harmonic oscillators under undirected topology as a special case finally numerical simulations are

lower upper implicit scheme for high speed inlet analysis on developpe une methode numerique d analyse du champ d ecoulement non visqueux d une entree a grande vitesse par resolution des equations d euler le schema se revele efficace et robuste aussi bien pour les faibles nombres de mach que pour les ecoulements transsoniques

lower upper implicit schemes with multiple grids for the euler equations on montre qu une methode de surrelaxation successive pour la resolution du schema implicite non factorise est equivalente a une forme modifiee du schema implicite inferieur superieur

synchronization of sampled data coupled harmonic oscillators with control inputs missing abstract this paper is concerned with the synchronization problem of sampled data coupled harmonic oscillators with control inputs missing a distributed algorithm for synchronization in undirected networks of coupled harmonic oscillators is proposed based on the sampled data measurement with controller failures some generic criteria for such algorithm over respectively undirected fixed and switching network topologies are derived analytically compared with some existing works a distinctive feature of this work is to solve synchronization problem in undirected networks even if each oscillator intermittently exchanges the velocity information with its neighbors in terms of sampled data setting only at discrete moments subsequently numerical examples illustrate and visualize the effectiveness and feasibility of the theoretical results

lower upper symmetric gauss seidel method for the euler and navier stokes equations on developpe un schema de relaxation multigrille application a des ecoulements transsoniques

second order leader following consensus of nonlinear multi agent systems via pinning control abstract without assuming that the interaction diagraph is strongly connected or contains a directed spanning tree this paper studies the second order leader following consensus problem of nonlinear multi agent systems with general network topologies based on graph theory matrix theory and lasalle s invariance principle a pinning control algorithm is proposed to achieve leader following consensus in a network of agents with nonlinear second order dynamics furthermore a pinning consensus protocol is developed for coupled double integrators with a constant reference velocity in particular this paper addresses what kind of agents and how many agents should be pinned and establishes some sufficient conditions to guarantee that all agents asymptotically follow the virtual leader numerical simulations are given to verify the theoretical analysis

sampled data synchronization control of dynamical networks with stochastic sampling this technical note is concerned with the sampled data synchronization control problem for a class of dynamical networks the sampling period considered here is assumed to be time varying that switches between two different values in a random way with given probability the addressed synchronization control problem is first formulated as an exponentially mean square stabilization problem for a new class of dynamical networks that involve both the multiple probabilistic interval delays mpids and the sector bounded nonlinearities sbns then a novel lyapunov functional is constructed to obtain sufficient conditions under which the dynamical network is exponentially mean square stable both gronwall s inequality and jenson integral inequality are utilized to substantially simplify the derivation of the main results subsequently a set of sampled data synchronization controllers is designed in terms of the solution to certain matrix inequalities that can be solved effectively by using available software finally a numerical simulation example is employed to show the effectiveness of the proposed sampled data synchronization control scheme

partially implicit scheme for chemically reacting flows at all mach numbers chemical reactions make computations of chemically reacting e ows stiff the degree of stiffness increases as the mach number decreases because the e ow timescale increases while the reaction timescales remain constant thus the computation of reacting e ows at low mach numbers is more dife cult than that in high mach number in the present study a new implicit scheme that employs partially implicit treatment of chemical source terms at mixed time levels has been developed the chemical jacobian is reduced to a partial form of the full chemical jacobian a lower triangular matrix and thus saves timein matrix inversion in addition robust calculation of the reacting e ows is made possible because a negative real eigenvalue of the partial chemical jacobian allows larger time step sizes than the full chemical jacobian for applications at all mach numbers a preconditioned lower upper symmetric gauss seidel scheme employing an approximate e ux jacobian splitting is incorporated for high mach number e ows the partially implicit scheme maintains similar convergence rates to the fully implicit one therefore it enhances computational efe ciency by reducing computing time per iteration for low machnumber reacting e ows it also enables robust calculation of reactions as well as the fully implicit one however its convergence rate is rather slow furthermore more stable and efe cient computations are possible when the fully implicit scheme is coupled with the partially implicit one

kinetic node pair formulation for two dimensional flows from continuum to transitional regime a hybrid finite element finite volume node pair discretization of conservation laws is reformulated in terms of a bhatnagar gross krook kinetic scheme to address flows from the continuum up to the transitional regime in a seamless fashion integrals of the particle distribution function from the kinetic theory of gases are adopted to compute the numerical fluxes along the boundary of each control volume flow features typical of the transitional regime like velocity and temperature slip condition at solid walls are automatically assured by the kinetic formulation of the node pair boundary conditions exemplary two dimensional numerical experiments ranging from continuum flows up to the transitional regime are presented

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

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

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

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

monotonicity of control volume methods robustness of numerical methods for multiphase flow problems in porous media is important for development of methods to be used in a wide range of applications here we discuss monotonicity for a simplified problem of single phase flow but where the simulation grids and media are allowed to be general posing challenges to control volume methods we discuss discrete formulations of the maximum principle and derive sufficient criteria for discrete monotonicity for arbitrary nine point control volume discretizations for conforming quadrilateral grids in 2d these criteria are less restrictive than the m matrix property it is shown that it is impossible to construct nine point methods which unconditionally satisfy the monotonicity criteria when the discretization satisfies local conservation and exact reproduction of linear potential fields numerical examples are presented which show the validity of the criteria for monotonicity further the impact of nonmonotonicity is studied different behavior for different discretization methods is illuminated and simple ideas are presented for improvement in terms of monotonicity

comparison of nonequilibrium solution algorithms applied to chemically stiff hypersonic flows three solution algorithms explicit under relaxation point implicit and lower upper symmetric gauss seidel are used to compute nonequilibrium flow around the apollo 4 return capsule at the 62 km altitude point in its descent trajectory by varying the mach number the efficiency and robustness of the solution algorithms were tested for different levels of chemical stiffness the performance of the solution algorithms degraded as the mach number and stiffness of the flow increased at mach 15 and 30 the lower upper symmetric gauss seidel method produces an eight order of magnitude drop in the energy residual in one third to one half the cray c 90 computer time as compared to the point implicit and explicit under relaxation methods the explicit under relaxation algorithm experienced convergence difficulties at mach 30 and above at mach 40 the performance of the lower upper symmetric gauss seidel algorithm deteriorates to the point that it is out performed by the point implicit method the effects of the viscous terms are investigated grid dependency questions are explored

discretization on unstructured grids for inhomogeneous anisotropic media part i derivation of the methods discretization methods are proposed for control volume formulations on polygonal and triangular grid cells in two space dimensions the methods are applicable for any system of conservation laws where the flow density is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy this paper gives a derivation of the methods and the relation to previously published methods is also discussed a further discussion of the methods including numerical examples is given in the companion paper part ii siam j sci comput pp 1717 1736

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

numerical navier stokes solutions from gas kinetic theory abstract in order to test the gas kinetic based hydrodynamic scheme given in paper i k h prendergast and k xu j comput phys 109 53 1993 as a navier stokes solver we extend the scheme to two dimensions and exhibit some navier stokes solutions the scheme is a high resolution gas kinetic scheme in both space and time the advective and diffusive fluxes are coupled and solved at the same time by following the time dependent velocity distribution function for particles numerical results for some well defined euler and navier stokes test cases are presented the kolmogorov and the laminar boundary layer problems exhibit the dominance of the real viscosity effects the strong shock interaction test cases show the applicability of the scheme to supersonic gas flow

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

synchronization of coupled harmonic oscillators with local instantaneous interaction in this brief we propose a distributed algorithm for synchronization of coupled harmonic oscillators with local instantaneous interaction we provide convergence analysis for such algorithm without and with a leader over fixed and switching undirected network topologies respectively we also establish the conditions by which the coupled harmonic oscillators achieve synchronized oscillatory motions under instantaneous network connectivity it is shown that the coupled harmonic oscillators can be synchronized even each oscillator instantaneously exchanges the velocity information with its neighbors only at discrete moments subsequently numerical examples illustrate and visualize the effectiveness and feasibility of the theoretical results

on the multidimensional gas kinetic bgk scheme this paper concerns the performance difference among the multidimensional md gas kinetic bgk scheme the corresponding quasi one dimensional q1d extension and the directional splitting ds scheme with kinetic boundary conditions the md scheme includes tangential slopes in the flux calculation which are absent in a q1d or ds method in spite of taking more computational time the md scheme is found to be able to capture the characteristics of rarefied flow having a boundary with curvature or a nonuniform temperature such as the inverted velocity distribution in rarefied cylindrical couette flow and the weak flow field induced by the temperature gradient of a body where the q1d fails the md scheme can also yield clearly better results in high speed microchannel flow and power law fluid flow between concentric rotating cylinders when compared with direct simulation monte carlo studies and analytic solutions in the low reynolds number flow around a naca0012 airfoil case the md scheme and the q1d method predict wall heat flux distribution with obvious difference the ds method predicts results nearly identical to those from q1d in these steady flow cases except that it can hardly give reasonable solutions in power law fluid case with viscous exponential factor away from unity correct prediction of the stress and the wall temperature gradient is considered responsible for this better performance in simulations of the scalar convection diffusion equation it is found that the inclusion of tangential slopes in the flux computation can clearly improve the temporal accuracy of the scheme in this case the ds scheme also shows good performance the present study suggests caution in adopting of the q1d extension or ds method in multidimensional flow when it is sensitive to the accuracy of stress or wall variable gradient calculations it is better to use the md scheme for high performance simulation

an immersed boundary method based on the gas kinetic bgk scheme for incompressible viscous flow in this paper an immersed boundary ib method based on the gas kinetic bhatnagar gross krook bgk scheme is proposed to simulate the incompressible viscous flow around stationary and moving rigid bodies in the presented ib method a set of lagrangian points represent the solid boundary and will exert force on the surrounding eulerian points which are the cell centers within the finite volume framework this force is calculated by a special iterative procedure and has an effect on the flux at the interface of cell which guarantees the fulfillment of the no slip boundary condition and entirely avoids the flow penetration the flow field is obtained from the bgk scheme with local grid refinement without complex mesh generation and transformation the present technique can be conveniently applied to simulations with complex and moving solid boundaries the second order temporal accuracy and better than first order spatial accuracy in l 2 norm are testified by the simulation of stokes first problem other three different test cases including flows around a stationary circular cylinder two circular cylinders in tandem and an oscillating circular cylinder demonstrate the good capability of the present method expanding the immersed boundary ib method to the finite volume gas kinetic bhatnagar gross krook bgk scheme the flux across the cell s interface will be influenced by the immersed boundary force the flow penetration to the immersed boundary is completely eliminated the present method can be conveniently applied to problems with moving objects

fully implicit chebyshev time spectral method for general unsteady flows an efficient fully implicit chebyshev time spectral method tsm is developed to solve general unsteady flows with or without periodicity high accuracy of the chebyshev time spectral operator is d

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

consensus networks with switching topology and time delays over finite fields the consensus problem in networks with both switching topology and time delays over finite fields is investigated in this paper the finite field which is a kind of finite alphabet is considered due to the fact that networks often possess limited computation memory and capabilities of communication first by graph theoretic method one necessary and sufficient condition is derived for finite field consensus of switching networks without time delays subsequently another necessary and sufficient condition on finite field consensus without time delays is provided based on ffc property of matrices associated with switching networks moreover by means of the results on delay free networks some necessary and sufficient conditions for finite field consensus of networks with both switching topology and time delays are obtained additionally it can be shown that switching networks with time delays present in each self transmission cannot achieve consensus

discretization on unstructured grids for inhomogeneous anisotropic media part ii discussion and numerical results a companion paper part i siam j sci comput 19 1998 pp 1700 1716 presents discretization methods for control volume formulations in two space dimensions properties of the developed methods are further discussed including rotational invariance and stability under the influence of anisotropy strong anisotropy may have to be handled by use of stretched grid cells numerical examples that demonstrate strengths and weaknesses of the methods are presented the examples focus on inhomogeneity anisotropy and rotational invariance

a reconstructed discontinuous galerkin method for incompressible flows on arbitrary grids abstract the discontinuous galerkin dg methods have attained increasing popularity for solving the incompressible navier stokes ins equations in recent years however the dg methods have their own weakness due to the high computational costs and storage requirements in this work we develop a high order hybrid reconstructed dg rdg method for solving the ins equations on arbitrary grids to be specific the inviscid term of the ins equations is discretized by applying the third order hybrid rdg p 1 p 2 method with a simplified artificial compressibility flux while the viscous term of the ins equations is discretized by using the simple direct dg ddg method a number of incompressible flow problems in both steady and unsteady forms for a variety flow conditions are computed to numerically assess the performance of the hybrid rdg p 1 p 2 method which confirm its ability to achieve the optimal third order of accuracy at a significantly reduced computational costs furthermore a detailed comparison of a variety of different reconstructed strategies is performed and presented numerical results demonstrate that the hybrid rdg p 1 p 2 method outperforms the rdg p 1 p 2 method based on either the original least squares reconstruction or the green gauss reconstruction for solving the ins equations

implicit gas kinetic unified algorithm based on multi block docking grid for multi body reentry flows covering all flow regimes based on the previous researches of the gas kinetic unified algorithm gkua for flows from highly rarefied free molecule transition to continuum a new implicit scheme of cell centered finite volume method is presented for directly solving the unified boltzmann model equation covering various flow regimes in view of the difficulty in generating the single block grid system with high quality for complex irregular bodies a multi block docking grid generation method is designed on the basis of data transmission between blocks and the data structure is constructed for processing arbitrary connection relations between blocks with high efficiency and reliability as a result the gas kinetic unified algorithm with the implicit scheme and multi block docking grid has been firstly established and used to solve the reentry flow problems around the multi bodies covering all flow regimes with the whole range of knudsen numbers from 10 to 3 7e6 the implicit and explicit schemes are applied to computing and analyzing the supersonic flows in near continuum and continuum regimes around a circular cylinder with careful comparison each other it is shown that the present algorithm and modelling possess much higher computational efficiency and faster converging properties the flow problems including two and three side by side cylinders are simulated from highly rarefied to near continuum flow regimes and the present computed results are found in good agreement with the related dsmc simulation and theoretical analysis solutions which verify the good accuracy and reliability of the present method it is observed that the spacing of the multi body is smaller the cylindrical throat obstruction is greater with the flow field of single body asymmetrical more obviously and the normal force coefficient bigger while in the near continuum transitional flow regime of near space flying surroundings the spacing of the multi body increases to six times of the diameter of the single body the interference effects of the multi bodies tend to be negligible the computing practice has confirmed that it is feasible for the present method to compute the aerodynamics and reveal flow mechanism around complex multi body vehicles covering all flow regimes from the gas kinetic point of view of solving the unified boltzmann model velocity distribution function equation

modeled boltzmann equation and its application to direct aeroacoustics simulation the bhatnagar gross and krook modeled boltzmann equation has been applied to simulate different fluid dynamics problems with varying degrees of success however its application to direct aeroacoustic computation is less successful one possible reason could be its inability to recover the state equation correctly for a diatomic gas and hence an inaccurate determination of the speed of sound the present study reports on the development of an improved modeled boltzmann equation for aeroacoustics simulation the approach is to modify the maxwellian distribution normally assumed for the equilibrium particle distribution function constraints imposed are the exact recovery of the state equation for a diatomic gas and the euler equations without invoking the small mach number assumption thus formulated a distribution function consisting of the maxwellian distribution plus three other terms that attempt to account for particle particle collisions is obtained a velocity lattice method is used to solve the improved modeled boltzmann equation using an equivalent lattice equilibrium distribution function the simulations are validated against benchmark aeroacoustic problems whose solutions are deduced from a direct numerical simulation of the euler equations the results of the improved modeled boltzmann equation obtained using a smaller computational domain are in excellent agreement with those deduced from direct numerical simulation using a larger computational domain thus verifying the viability and correctness of the modified equilibrium distribution function

synchronization of coupled harmonic oscillators with local interaction this paper studies synchronization of coupled second order linear harmonic oscillators with local interaction we analyze convergence conditions over respectively directed fixed and switching network topologies by using tools from algebraic graph theory matrix theory and nonsmooth analysis it is shown that the coupled harmonic oscillators can be synchronized under mild network connectivity conditions examples are given to validate the convergence conditions the theoretical result is also applied to synchronized motion coordination of multi agent systems as a proof of concept

implicit boundary equations for conservative navier stokes equations abstract co existence of the physical and numerical boundary conditions makes implicit boundary treatment a particularly difficult problem in modern cfd simulations previous studies adopted space time extrapolation or specially designed partial differential equations on the boundaries that are different from those of interior points they are often formulated in terms of primitive variables and are very challenging for complicated boundary types to be converted to and implemented in the conservative variables that are preferred in numerical simulations more importantly different boundary equations or different extrapolation techniques may compromise the stability accuracy or convergence rate of the a stable schemes that are developed for interior points a new methodology for implicit boundary treatment is proposed in this study by introducing a simple correction matrix t a set of generalized equations q t i t r are developed in terms of conservative variables it is applicable for both the interior domain t 0 and the boundaries t 0 it is in a partial differential equation form satisfies the boundary conditions accurately independent of the time and spatial discretizations any one sided schemes can be used on the boundaries but still maintain the upwind property implicit solution techniques are made significantly easy to implement using for example the data parallel lower upper relation method and the newton method combined with the gmres method for subsidiary iterations numerical experiments show that the proposed methodology produces stable simulations for very large cfl numbers and preserve the imposed boundary values accurately

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

a robust low diffusive kinetic scheme for the navier stokes euler equations a new kinetic scheme based on the equilibrium flux method efm and modified using osher intermediate states is proposed this new scheme called efmo combines the robustness of the equilibrium flux method and the accuracy of flux difference splitting schemes the original efm scheme is expressed in terms of simple wave decomposition in which only the linearly degenerate subpath is calculated from osher numerical flux while nonlinear waves are still evaluated from the regular efm splitting owing to its capability of withstanding intense nonlinear waves and yet exactly resolving contact discontinuities efmo is particularly well suited for the resolution of the navier stokes equations as demonstrated by a series of severe test cases including the high speed viscous flow around a cone a shock boundary layer interaction problem a vacuum apparition problem the hypersonic flow around a circular cylinder at mach 100 and the forward facing step at mach 3

decoupled implicit method for aerothermodynamics and reacting flows we propose a new implicit computational fluid dynamics method for steady state compressible reacting flows the concept is to decouple the total mass momentum and energy conservation equations from the species mass and internal energy equations and to solve the two equation sets sequentially with certain approximations to the implicit system it is possible to dramatically reduce the cost of the solution with little to no penalty on the convergence properties importantly the cost of the decoupled implicit problem scales linearly with the number of species as opposed to the quadratic scaling for the conventional fully coupled method furthermore the new approach reduces the memory requirements by a significant factor the decoupled implicit method shows promise for the application to aerothermodynamics problems and reacting flows

robust convergence of multi point flux approximation on rough grids this paper establishes the convergence of a multi point flux approximation control volume method on rough quadrilateral grids by rough grids we refer to a family of refined quadrilateral grids where the cells are not required to approach parallelograms in the asymptotic limit in contrast to previous convergence results for these methods we consider here a version where the flux approximation is derived directly in the physical space and not on a reference cell as a consequence less regular grids are allowed however the extra cost is that the symmetry of the method is lost

implicit navier stokes solver for three dimensional compressible flows a three dimensional numerical method based on the lower upper symmetric gauss seidel implicit scheme in conjunction with the flux limited dissipation model is developed for solving the compressible navier stokes equations a new computer code based on this method requires only 9 s per grid point per iteration on a single processor of a cray ymp computer and executes at the sustained rate of 175 mflops a reduction of three orders of magnitude in the residual for a high reynolds number flow using 636k grid points is obtained in 38 min

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

implicit treatment of diffusion terms in lower upper algorithms a method is presented which allows diffusion terms to be treated implicitly in the lower upper lu algorithm which is a commonly used method for solving compressible euler and navier stokes equations so that the algorithm s good stability properties will not be impaired the new method generalizes the concept of lu factorization from that associated with the sign of eigenvalues to that associated with backward and forward difference operators without regard to eigenvalues the method is verified in a turbulent boundary layer study

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