spectral lagrangian methods for collisional models of non equilibrium statistical states we propose a new spectral lagrangian based deterministic solver for the non linear boltzmann transport equation bte in d dimensions for variable hard sphere vhs collision kernels with conservative or non conservative binary interactions the method is based on symmetries of the fourier transform of the collision integral where the complexity in its computation is reduced to a separate integral over the unit sphere s d 1 the conservation of moments is enforced by lagrangian constraints the resulting scheme implemented in free space is very versatile and adjusts in a very simple manner to several cases that involve energy dissipation due to local micro reversibility inelastic interactions or elastic models of slowing down process our simulations are benchmarked with available exact self similar solutions exact moment equations and analytical estimates for the homogeneous boltzmann equation both for elastic and inelastic vhs interactions benchmarking of the simulations involves the selection of a time self similar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for maxwell molecules as studied first in bobylev et al a v bobylev c cercignani g toscani proof of an asymptotic property of self similar solutions of the boltzmann equation for granular materials journal of statistical physics 111 2003 403 417 and generalized to a wide range of related models in bobylev et al a v bobylev c cercignani i m gamba on the self similar asymptotics for generalized non linear kinetic maxwell models communication in mathematical physics in press url the method also produces accurate results in the case of inelastic diffusive boltzmann equations for hard spheres inelastic collisions under thermal bath where overpopulated non gaussian exponential tails have been conjectured in computations by stochastic methods t v noije m ernst velocity distributions in homogeneously cooling and heated granular fluids granular matter 1 57 1998 m h ernst r brito scaling solutions of inelastic boltzmann equations with over populated high energy tails journal of statistical physics 109 2002 407 432 s j moon m d shattuck j swift velocity distributions and correlations in homogeneously heated granular media physical review e 64 2001 031303 i m gamba s rjasanow w wagner direct simulation of the uniformly heated granular boltzmann equation mathematical and computer modelling 42 2005 683 700 and rigorously proven in gamba et al i m gamba v panferov c villani on the boltzmann equation for diffusively excited granular media communications in mathematical physics 246 2004 503 541 39 and a v bobylev i m gamba v panferov moment inequalities and high energy tails for boltzmann equations with inelastic interactions journal of statistical physics 116 2004 1651 1682

symplectic conditions and stochastic generating functions of stochastic runge kutta methods for stochastic hamiltonian systems with multiplicative noise abstract this paper develops numerical approximate algorithms for the solutions of stochastic hamiltonian systems with multiplicative noise some conditions are captured to guarantee that a given stochastic runge kutta method is symplectic this paper also shows that stochastic symplectic runge kutta methods can be written in terms of stochastic generating functions based on the order and symplectic conditions some low stage stochastic symplectic runge kutta methods with strong global order 1 0 are constructed some numerical results are presented to demonstrate the efficiency of stochastic symplectic runge kutta methods

efficient asymptotic preserving ap schemes for some multiscale kinetic equations many kinetic models of the boltzmann equation have a diffusive scaling that leads to the navier stokes type parabolic equations as the small scaling parameter approaches zero in practical applications it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter from the rarefied kinetic regimes to the hydrodynamic diffusive regimes an earlier approach in s jin l pareschi and g toscani siam j numer anal 35 1998 pp 2405 2439 reformulates such systems into the common hyperbolic relaxation system by jin and xin for hyperbolic conservation laws used to construct the relaxation schemes and then uses a multistep time splitting method to solve the relaxation system here we observe that the combination of the two time split steps may yield hyperbolic parabolic systems that are more advantageous in both stability and efficiency for numerical computations we show that such an approach yields a class of asymptotic preserving ap schemes which are suitable for the computation of multiscale kinetic problems we use the goldstein taylor and carleman models to illustrate this approach

on the asymptotic preserving property of the unified gas kinetic scheme for the diffusion limit of linear kinetic models the unified gas kinetic scheme ugks of k xu et al 2010 37 originally developed for multiscale gas dynamics problems is applied in this paper to a linear kinetic model of radiative transfer theory while such problems exhibit purely diffusive behavior in the optically thick or small knudsen regime we prove that ugks is still asymptotic preserving ap in this regime but for the free transport regime as well moreover this scheme is modified to include a time implicit discretization of the limit diffusion equation and to correctly capture the solution in case of boundary layers contrary to many ap schemes this method is based on a standard finite volume approach it does neither use any decomposition of the solution nor staggered grids several numerical tests demonstrate the properties of the scheme

conservative stochastic differential equations mathematical and numerical analysis we consider stochastic differential equations on the whole euclidean space possessing a scalar invariant along their solutions the stochastic dynamics therefore evolves on a hypersurface of the ambient space using orthogonal coordinate systems we show the existence and uniqueness of smooth solutions of the kolmogorov equation under some ellipticity conditions over the invariant hypersurfaces if we assume moreover the existence of an invariant measure we show the exponential convergence of the solution towards its average in the second part we consider numerical approximation of the stochastic differential equation and show the convergence and numerical ergodicity of a class of projected schemes in the context of molecular dynamics this yields numerical schemes that are ergodic with respect to the microcanonical measure over isoenergy surfaces

management of discontinuous reconstruction in kinetic schemes the present paper highlights the importance of management of the discontinuous reconstruction in the kinetic schemes for gasdynamic equation systems firstly it is revealed by the analysis of the gas kinetic bgk scheme jcp 171 2001 289 that a continuous reconstruction created from a discontinuous one is a key to the successful kinetic schemes when it is applied to a well resolved region the numerical flux that takes account of the collision effect correctly becomes lax wendroff like when it is applied to an unresolved region such as a shock layer an appreciable numerical dissipation which contributes to the suppression of spurious oscillations is produced secondly new kinetic schemes for the compressible navier stokes euler equations are developed by using the key the numerical flux of one of the schemes is computed by using the splitting algorithm where the effect of the molecular collision is directly taken into account and the undesirable error of the splitting algorithm in the case where the time step is much larger than the mean free time is avoided by a simple modification of the initial data although a discontinuous reconstruction is employed in the approximation of the initial data the continuity is automatically taken into account in the dominant part of the numerical flux the other schemes are the extensions of the lax wendroff type scheme to the case of the key reconstruction thirdly the performance of these schemes is tested it is demonstrated that they work as shock capturing schemes and yield fine boundary layer profiles with a reasonable number of cells such as 10 cells in the layer

a class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources in this paper we propose a general time discrete framework to design asymptotic preserving schemes for initial value problem of the boltzmann kinetic and related equations numerically solving these equations are challenging due to the nonlinear stiff collision source terms induced by small mean free or relaxation time we propose to penalize the nonlinear collision term by a bgk type relaxation term which can be solved explicitly even if discretized implicitly in time moreover the bgk type relaxation operator helps to drive the density distribution toward the local maxwellian thus naturally imposes an asymptotic preserving scheme in the euler limit the scheme so designed does not need any nonlinear iterative solver or the use of wild sum it is uniformly stable in terms of the possibly small knudsen number and can capture the macroscopic fluid dynamic euler limit even if the small scale determined by the knudsen number is not numerically resolved it is also consistent to the compressible navier stokes equations if the viscosity and heat conductivity are numerically resolved the method is applicable to many other related problems such as hyperbolic systems with stiff relaxation and high order parabolic equations

numerical methods for stochastic systems preserving symplectic structure stochastic hamiltonian systems with multiplicative noise phase flows of which preserve symplectic structure are considered to construct symplectic methods for such systems sufficiently general fully implicit schemes i e schemes with implicitness both in deterministic and stochastic terms are needed a new class of fully implicit methods for stochastic systems is proposed increments of wiener processes in these fully implicit schemes are substituted by some truncated random variables a number of symplectic integrators is constructed special attention is paid to systems with separable hamiltonians some results of numerical experiments are presented they demonstrate superiority of the proposed symplectic methods over very long times in comparison with nonsymplectic ones

stochastic symplectic partitioned runge kutta methods for stochastic hamiltonian systems with multiplicative noise a new class of sprk methods and their order conditions are given symplectic conditions of the sprk methods are obtained stochastic generating functions of the ssprk methods are obtained it is proved that the ssprk methods can preserve the quadratic invariants some low stage ssprk methods are constructed some new stochastic partitioned runge kutta sprk methods are proposed for the strong approximation of partitioned stochastic differential equations sdes the order conditions up to strong global order 1 0 are calculated the sprk methods are applied to solve stochastic hamiltonian systems with multiplicative noise some conditions are captured to guarantee that a given sprk method is symplectic it is shown that stochastic symplectic partitioned runge kutta ssprk methods can be written in terms of stochastic generating functions in addition this paper also proves that the ssprk methods can conserve the quadratic invariants of original stochastic systems based on the order and symplectic conditions some low stage ssprk methods with strong global order 1 0 are constructed finally some numerical results are presented to demonstrate the efficiency of the ssprk methods

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

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

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

uniformly stable numerical schemes for the boltzmann equation preserving the compressible navier stokes asymptotics in this paper we develop a numerical method to solve boltzmann like equations of kinetic theory which is able to capture the compressible navier stokes dynamics at small knudsen numbers our approach is based on the micro macro decomposition technique which applies to general collision operators this decomposition is performed in all the phase space and leads to an equivalent formulation of the boltzmann or bgk equation that couples a kinetic equation with macroscopic ones this new formulation is then discretized with a semi implicit time scheme combined with a staggered grid space discretization finally several numerical tests are presented in order to illustrate the efficiency of our approach incidentally we also introduce in this paper a modification of a standard splitting method that allows to preserve the compressible navier stokes asymptotics in the case of the simplified bgk model up to our knowledge this property is not known for general collision operators

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

balanced implicit methods for stiff stochastic systems this paper introduces some implicitness in stochastic terms of numerical methods for solving stiff stochastic differential equations and especially a class of fully implicit methods the balanced methods their order of strong convergence is proved numerical experiments compare the stability properties of these schemes with explicit ones

numerical simulations for stochastic differential equations on manifolds by stochastic symmetric projection method abstract stochastic standard projection technique as an efficient approach to simulate stochastic differential equations on manifolds is widely used in practical applications however stochastic standard projection methods usually destroy the geometric properties such as symplecticity or reversibility even though the underlying methods are symplectic or symmetric which seriously affect long time behavior of the numerical solutions in this paper a modification of stochastic standard projection methods for stochastic differential equations on manifolds is presented the modified methods called the stochastic symmetric projection methods remain the symmetry and the reversibility of the underlying methods and maintain the numerical solutions on the correct manifolds the mean square convergence order of these methods are proved to be the same as the underlying methods numerical experiments are implemented to verify the theoretical results and show the superiority of the stochastic symmetric projection methods over the stochastic standard projection methods

a class of orthogonal integrators for stochastic differential equations the purpose of this paper is to construct a class of orthogonal integrators for stochastic differential equations sdes the family of sdes with orthogonal solutions is univocally characterized for this a class of orthogonal integrators is introduced by imposing constraints to runge kutta rk matrices and weights of the standard stochastic rk schemes the performance of the method is illustrated by means of numerical simulations

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

numerical passage from kinetic to fluid equations a numerical way to pass from the bhatnagar gross krook model of the boltzmann equation to compressible euler equations is presented in order to do so a stable discretization of the kinetic equati

numerical simulation of flows from free molecular regime to continuum regime by a dvm with streaming and collision processes a discrete velocity method dvm with streaming and collision processes is presented in this work for simulation of flows from free molecular regime to continuum regime the present scheme can be considered as a semi lagrangian like scheme at first we follow the conventional dvm to discretize the phase velocity space by a number of discrete velocities then for each discrete velocity the kinetic equation with bgk shakhov model is integrated in space and time within one time step as a result a simple algebraic formulation can be obtained and its solution can be marched in time by the streaming and collision processes however differently from the conventional semi lagrangian scheme the present scheme uses the muscl approach with van albada limiter in the process of reconstructing the distribution function at the surrounding points of the cell center and the transport distance is controlled in order to avoid extrapolation this makes the present scheme be capable of simulating the hypersonic rarefied flows in addition as compared to the unified gas kinetic scheme ugks the present scheme is simpler and easier for implementation thus the computational efficiency can be improved accordingly to validate the proposed numerical scheme test examples from free molecular regime to continuum regime are simulated numerical results showed that the present scheme can predict the flow properties accurately even for hypersonic rarefied flows

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

microscopically implicit macroscopically explicit schemes for the bgk equation in this work a new class of numerical methods for the bgk model of kinetic equations is introduced the schemes proposed are implicit with respect to the distribution function while the macroscopic moments are evolved explicitly in this fashion the stability condition on the time step coincides with a macroscopic cfl evaluated using estimated values for the macroscopic velocity and sound speed thus the stability restriction does not depend on the relaxation time and it does not depend on the microscopic velocity of energetic particles either with the technique proposed here the updating of the distribution function requires the solution of a linear system of equations even though the bgk model is highly non linear thus the proposed schemes are particularly effective for high or moderate mach numbers where the macroscopic cfl condition is comparable to accuracy requirements we show results for schemes of order 1 and 2 and the generalization to higher order is sketched

numerical methods for stochastic delay differential equations via the wong zakai approximation we use the wong zakai approximation as an intermediate step to derive numerical schemes for stochastic delay differential equations by approximating the brownian motion with its truncated spectral expansion and then using different discretizations in time we present three schemes a predictor corrector scheme a midpoint scheme and a milstein like scheme we prove that the predictor corrector scheme converges with order half in the mean square sense while the milstein like scheme converges with order one numerical tests confirm the theoretical prediction and demonstrate that the midpoint scheme is of half order convergence numerical results also show that the predictor corrector and midpoint schemes can be of first order convergence under commutative noises when there is no delay in the diffusion coefficients

variational principles for stochastic soliton dynamics we develop a variational method of deriving stochastic partial differential equations whose solutions follow the flow of a stochastic vector field as an example in one spatial dimension we numerically simulate singular solutions peakons of the stochastically perturbed camassa holm ch equation derived using this method these numerical simulations show that peakon soliton solutions of the stochastically perturbed ch equation persist and provide an interesting laboratory for investigating the sensitivity and accuracy of adding stochasticity to finite dimensional solutions of stochastic partial differential equations in particular some choices of stochastic perturbations of the peakon dynamics by wiener noise canonical hamiltonian stochastic deformations ch sd allow peakons to interpenetrate and exchange order on the real line in overtaking collisions although this behaviour does not occur for other choices of stochastic perturbations which preserve the euler poincare structure of the ch equation parametric stochastic deformations p sd and it also does not occur for peakon solutions of the unperturbed deterministic ch equation the discussion raises issues about the science of stochastic deformations of finite dimensional approximations of evolutionary partial differential equation and the sensitivity of the resulting solutions to the choices made in stochastic modelling

weak first or second order implicit runge kutta methods for stochastic differential equations with a scalar wiener process new fully implicit stochastic runge kutta schemes of weak order 1 or 2 are proposed for stochastic differential equations with sufficiently smooth drift and diffusion coefficients and a scalar wiener process which are derivative free and which are a stable in mean square for a linear test equation in some general settings they are sought in a transparent way and their convergence order and stability properties are confirmed in numerical experiments

numerical methods for strong solutions of stochastic differential equations an overview this paper gives a review of recent progress in the design of numerical methods for computing the trajectories sample paths of solutions to stochastic differential equations we give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important with a particular focus on computational biology applications and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes we present the stochastic taylor series expansion as the fundamental mechanism for constructing effective numerical methods give general results that relate local and global order of convergence and mention the magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem we also present various classes of explicit and implicit methods for strong solutions based on the underlying structure of the problem finally we discuss implementation issues relating to maintaining the brownian path efficient simulation of stochastic integrals and variable step size implementations based on various types of control

deterministic solution of the spatially homogeneous boltzmann equation using discontinuous galerkin discretizations in the velocity space abstract we present a new deterministic approach for the solution of the spatially homogeneous boltzmann kinetic equation based on nodal discontinuous galerkin dg discretizations in the velocity space in the new approach the collision operator has the form of a bilinear operator with a pre computed kernel its evaluation requires o n 5 operations at every point of the phase space where n is the number of degrees of freedom in one velocity dimension the method is generalized to any molecular potential results of numerical simulations are presented for the problem of spatially homogeneous relaxation for the hard spheres potential comparison with the method of direct simulation monte carlo showed excellent agreement

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

explicit pseudo symplectic methods based on generating functions for stochastic hamiltonian systems abstract we propose a systematic approach to construct explicit pseudo symplectic schemes for stochastic hamiltonian systems this method is based on generating functions so it is an extension of the techniques used for constructing high order symplectic schemes for stochastic hamiltonian systems we study the order of convergence of the proposed explicit pseudo symplectic schemes the excellent long term performance of the pseudo symplectic schemes is verified numerically

high order stochastic symplectic partitioned runge kutta methods for stochastic hamiltonian systems with additive noise abstract in this paper a simple class of stochastic partitioned runge kutta sprk methods is proposed for solving stochastic hamiltonian systems with additive noise firstly the order conditions and symplectic condictions are analysised by using colored rooted tree theory then a family of mean square order 1 5 diagonally implicit stochastic symplectic partitioned runge kutta ssprk methods is presented moreover several explicit ssprk methods are constructed for systems with a separable hamiltonian h 0 v p u t q furthermore these methods are proved to converge with mean square order 2 0 to the solution when they are applied to second order stochastic hamiltonian systems with a separable hamiltonian and additive noise finally several numerical examples are performed to demonstrate efficiency of those ssprk methods

a resolution principle for constrained logics abstract we introduce a constrained logic scheme with a resolution principle for clauses whose variables are constrained by a constraint theory constraints can be seen as quantifier restrictions filtering out the values that any interpretation of the underlying constraint theory can assign to the variables of a formula with such restricted quantifiers we present a resolution principle for constrained clauses where unification is replaced by testing constraints for satisfiability over the constraint theory we show that constrained resolution is sound and complete in that a set of constrained clauses is unsatisfiable over the constraint theory if and only if for each model of the constraint theory we can deduce a constrained empty clause whose constraint is satisfiable in that model we demonstrate that we cannot require a better result in general but we give conditions under which at most finitely many such empty clauses are needed or even better only one empty clause as in classical resolution sorted resolution or resolution with theory unification

high order conformal symplectic and ergodic schemes for the stochastic langevin equation via generating functions in this paper we consider the stochastic langevin equation with additive noises which possesses both conformal symplectic geometric structure and ergodicity we propose a methodology of constructing high weak order conformal symplectic schemes by converting the equation into an equivalent autonomous stochastic hamiltonian system and modifying the associated generating function to illustrate this approach we construct a specific second order numerical scheme and prove that its symplectic form dissipates exponentially moreover for the linear case the proposed scheme is also shown to inherit the ergodicity of the original system and the temporal average of the numerical solution is a proper approximation of the ergodic limit over long time numerical experiments are given to verify these theoretical results

an asymptotic preserving two dimensional staggered grid method for multiscale transport equations we propose a two dimensional asymptotic preserving scheme for linear transport equations with diffusive scalings it is an extension of the time splitting developed by jin pareschi and toscani siam j numer anal 38 2000 pp 913 936 but uses spatial discretizations on staggered grids which preserves the discrete diffusion limit with a more compact stencil the first novelty of this paper is that we propose a staggering in two dimensions that requires fewer unknowns than one could have naively expected the second contribution of this paper is that we rigorously analyze the scheme of jin pareschi and toscani we show that the scheme is ap and obtain an explicit cfl condition which couples a hyperbolic and a parabolic condition this type of condition is common for asymptotic preserving schemes and guarantees uniform stability with respect to the mean free path in addition we obtain an upper bound on the relaxation parameter which is the crucial parameter of the used time discretization se

hamiltonian systems with levy noise symplecticity hamilton s principle and averaging principle abstract this work focuses on topics related to hamiltonian stochastic differential equations with levy noise we first show that the phase flow of the stochastic system preserves symplectic structure and propose a stochastic version of hamilton s principle by the corresponding formulation of the stochastic action integral and the euler lagrange equation based on these properties we further investigate the effective behavior of a small transversal perturbation to a completely integrable stochastic hamiltonian system with levy noise we establish an averaging principle in the sense that the action component of solution converges to the solution of a stochastic differential equation when the scale parameter goes to zero furthermore we obtain the estimation for the rate of this convergence finally we present an example to illustrate these results

exponential runge kutta for the inhomogeneous boltzmann equations with high order of accuracy we consider the development of exponential methods for the robust time discretization of space inhomogeneous boltzmann equations in stiff regimes compared to the space homogeneous case or more in general to the case of splitting based methods studied in dimarco pareschi 7 a major difficulty is that the local maxwellian equilibrium state change with respect to time and thus needs a proper numerical treatment we show how to derive asymptotic preserving ap schemes of arbitrary order and in particular by using the shu osher representation of runge kutta methods we explore the monotonicity properties of such schemes like strong stability preserving ssp and positivity preserving several numerical results confirm our analysis

unions of non disjoint theories and combinations of satisfiability procedures in this paper we outline a theoretical framework for the combination of decision procedures for constraint satisfiability we describe a general combination method which given a procedure that decides constraint satisfiability with respect to a constraint theory t1 and one that decides constraint satisfiability with respect to a constraint theory t2 produces a procedure that semi decides constraint satisfiability with respect to the union of t1 and t2 we provide a number of model theoretic conditions on the constraint language and the component constraint theories for the method to be sound and complete with special emphasis on the case in which the signatures of the component theories are non disjoint we also describe some general classes of theories to which our combination results apply and relate our approach to some of the existing combination methods in the field

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

a third order gas kinetic cpr method for the euler and navier stokes equations on triangular meshes abstract a third order accurate gas kinetic scheme based on the correction procedure via reconstruction cpr framework is developed for the euler and navier stokes equations on triangular meshes the scheme combines the accuracy and efficiency of the cpr formulation with the multidimensional characteristics and robustness of the gas kinetic flux solver comparing with high order finite volume gas kinetic methods the current scheme is more compact and efficient by avoiding wide stencils on unstructured meshes unlike the traditional cpr method where the inviscid and viscous terms are treated differently the inviscid and viscous fluxes in the current scheme are coupled and computed uniformly through the kinetic evolution model in addition the present scheme adopts a fully coupled spatial and temporal gas distribution function for the flux evaluation achieving high order accuracy in both space and time within a single step numerical tests with a wide range of flow problems from nearly incompressible to supersonic flows with strong shocks for both inviscid and viscous problems demonstrate the high accuracy and efficiency of the present scheme

a hybrid particle continuum method applied to shock waves a hybrid numerical scheme designed for hypersonic non equilibrium flows is presented which solves the navier stokes equations in regions of near equilibrium and uses the direct simulation monte carlo method where the flow is in non equilibrium detailed analysis of each stage of the hybrid cycle illustrates the difficulty in defining physically correct dsmc boundary conditions in regards to both macroscopic state and velocity distribution however results also show that dsmc boundary conditions have little effect on a previously initialized interior particle domain a sub relaxation technique capable of determining macroscopic hydrodynamic properties in a dsmc simulation is used to determine low scatter boundary conditions for the ns domain particle and continuum domains adapt during the hybrid simulation through application of a continuum breakdown parameter based on the gradient length knudsen number the hybrid code reproduces experimental results and full dsmc simulations in half the time for a large range of 1d shock waves in argon and diatomic nitrogen gas

a modular particle continuum numerical method for hypersonic non equilibrium gas flows a modular particle continuum mpc numerical method for steady state flows is presented which solves the navier stokes equations in regions of near equilibrium and uses the direct simulation monte carlo dsmc method to simulate regions of non equilibrium gas flow existing state of the art dsmc and navier stokes solvers are coupled together using a novel modular implementation which requires only a limited number of additional hybrid functions hybrid functions are used to adaptively position particle continuum interfaces and update boundary conditions in each module at appropriate times the mpc method is validated for 2d flow over a cylinder at various hypersonic mach numbers where the global knudsen number is 0 01 for the cases considered the mpc method is verified to accurately reproduce dsmc flow field results as well as local particle velocity distributions up to 2 2 times faster than full dsmc simulations

a new combination procedure for the word problem that generalizes fusion decidability results in modal logics previous results for combining decision procedures for the word problem in the non disjoint case do not apply to equational theories induced by modal logics which are not disjoint for sharing the theory of boolean algebras conversely decidability results for the fusion of modal logics are strongly tailored towards the special theories at hand and thus do not generalize to other types of equational theories in this paper we present a new approach for combining decision procedures for the word problem in the non disjoint case that applies to equational theories induced by modal logics but is not restricted to them the known fusion decidability results for modal logics are instances of our approach however even for equational theories induced by modal logics our results are more general since they are not restricted to so called normal modal logics

exponential runge kutta methods for stiff kinetic equations we introduce a class of exponential runge kutta integration methods for kinetic equations the methods are based on a decomposition of the collision operator into an equilibrium and a nonequilibrium part and are exact for relaxation operators of bgk type for boltzmann type kinetic equations they work uniformly for a wide range of relaxation times and avoid the solution of nonlinear systems of equations even in stiff regimes we give sufficient conditions in order that such methods are unconditionally asymptotically stable and asymptotic preserving such stability properties are essential to guarantee the correct asymptotic behavior for small relaxation times the methods also offer favorable properties such as nonnegativity of the solution and entropy inequality for this reason as we will show the methods are suitable both for deterministic as well as probabilistic numerical techniques

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

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

numerical solution of the boltzmann equation i spectrally accurate approximation of the collision operator in this paper we show that the use of spectral galerkin methods for the approximation of the boltzmann equation in the velocity space permits one to obtain spectrally accurate numerical solutions at a reduced computational cost we prove that the spectral algorithm preserves the total mass and approximates with infinite order accuracy momentum and energy consistency of the method is also proved and a stability result for a smoothed positive scheme is given we demonstrate that the fourier coefficients associated with the collision kernel of the equation have a very simple structure and in some cases can be computed explicitly numerical examples for homogeneous test problems in two and three dimensions confirm the advantages of the method

high weak order methods for stochastic differential equations based on modified equations inspired by recent advances in the theory of modified differential equations we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations this approach is illustrated with the constructions of new methods of weak order two in particular semi implicit integrators well suited for stiff mean square stable stochastic problems and implicit integrators that exactly conserve all quadratic first integrals of a stochastic dynamical system numerical examples confirm the theoretical results and show the versatility of our methodology

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

computing ergodic limits for langevin equations abstract to evaluate averages with respect to the invariant law for langevin type equations one has to integrate a system over long time intervals especially when dissipation is small this is a challenging problem from the computational point of view since coefficients of the langevin equations are typically not globally lipschitz the difficulties are redoubled in the nonglobal lipschitz case we observe an exploding behavior of some approximate trajectories due to many usual numerical methods in g n milstein m v tretyakov numerical integration of stochastic differential equations with nonglobally lipschitz coefficients siam j numer anal 43 2005 1139 1154 we propose and justify a concept which in principle allows us to apply any numerical method to stochastic differential equations with nonglobally lipschitz coefficients here quasi symplectic integrators from g n milstein m v tretyakov quasi symplectic methods for langevin type equations ima j numer anal 23 2003 593 626 they are the most appropriate methods for solving langevin type equations especially when damping is small and the concept of g n milstein m v tretyakov numerical integration of stochastic differential equations with nonglobally lipschitz coefficients siam j numer anal 43 2005 1139 1154 are applied for calculation of ergodic limits special attention is paid to the case when the invariant measure is known e g when this measure is gibbsian some computer experiments with three model problems van der pol s equation with additive noise one dimensional arrays of oscillators in thermal equilibrium and a physical pendulum with linear friction and additive noise are presented

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

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

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

efficient strong integrators for linear stochastic systems we present numerical schemes for the strong solution of linear stochastic differential equations driven by an arbitrary number of wiener processes these schemes are based on the neumann stochastic taylor and magnus expansions first we consider the case when the governing linear diffusion vector fields commute with each other but not with the linear drift vector field we prove that numerical methods based on the magnus expansion are more accurate in the mean square sense than corresponding stochastic taylor integration schemes second we derive the maximal rate of convergence for arbitrary multidimensional stochastic integrals approximated by their conditional expectations consequently for general nonlinear stochastic differential equations with noncommuting vector fields we deduce explicit formulae for the relation between error and computational costs for methods of arbitrary order third we consider the consequences in two numerical studies one of which is an application arising in stochastic linear quadratic optimal control

stochastic symplectic methods based on the pade approximations for linear stochastic hamiltonian systems in this article we propose a kind of numerical methods based on the pade approximations for two kinds of stochastic hamiltonian systems for the general linear stochastic hamiltonian systems it is shown that the applied pade approximations p k k produce numerical solutions that are symplectic and the proposed numerical schemes based on p r s are of root mean square convergence order r s 2 for a special kind of linear stochastic hamiltonian systems with additive noises the numerical methods using two kinds of pade approximations p r s and p r 1 possess root mean square convergence order r 2 when r s r 3 and are symplectic if r s these generalize the pade approximation approaches for symplectic integration of linear hamiltonian systems to the stochastic context

discrete gradient approach to stochastic differential equations with a conserved quantity general autonomous stochastic differential equations sdes driven by one dimensional brownian motion in the stratonovich sense with a conserved quantity i y are considered relying on this conserved quantity an equivalent skew gradient sg form of original sdes is constructed with the aim of constructing conserved numerical methods direct discrete gradient approaches which approximate directly the sg system lead to two conserved methods in indirect discrete gradient approaches we first split the sg system into subsystems which preserve the original conserved quantity then apply direct discrete gradient approaches to subsystems to produce two composition schemes the mean square convergence of order 1 for these four methods depends on the assumptions that the coefficients of sdes are lipschitz continuous with bounded second moments along the solution of sdes are twice continuous differentiable and have derivatives that are uniformly bounded three numerical experiments are presented to verify our theoretical analysis and show the advantages of these methods

accuracy analysis of high order lattice boltzmann models for rarefied gas flows in this work we have theoretically analyzed and numerically evaluated the accuracy of high order lattice boltzmann lb models for capturing non equilibrium effects in rarefied gas flows in the incompressible limit the lb equation is shown to be able to reduce to the linearized bhatnagar gross krook bgk equation therefore when the same gauss hermite quadrature is used lb method closely resembles the discrete velocity method dvm in addition the order of hermite expansion for the equilibrium distribution function is found not to be directly correlated with the approximation order in terms of the knudsen number to the bgk equation for incompressible flows meanwhile we have numerically evaluated the lb models for a standing shear wave problem which is designed specifically for assessing model accuracy by excluding the influence of gas molecule surface interactions at wall boundaries the numerical simulation results confirm that the high order terms in the discrete equilibrium distribution function play a negligible role in capturing non equilibrium effect for low speed flows by contrast appropriate gauss hermite quadrature has the most significant effect on whether lb models can describe the essential flow physics of rarefied gas accurately our simulation results where the effect of wall gas interactions is excluded can lead to conclusion on the lb modeling capability that the models with higher order quadratures provide more accurate results for the same order gauss hermite quadrature the exact abscissae will also modestly influence numerical accuracy using the same gauss hermite quadrature the numerical results of both lb and dvm methods are in excellent agreement for flows across a broad range of the knudsen numbers which confirms that the lb simulation is similar to the dvm process therefore lb method can offer flexible models suitable for simulating continuum flows at the navier stokes level and rarefied gas flows at the linearized boltzmann model equation level

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

b series analysis of stochastic runge kutta methods that use an iterative scheme to compute their internal stage values in recent years implicit stochastic runge kutta srk methods have been developed both for strong and weak approximations for these methods the stage values are only given implicitly however in practice these implicit equations are solved by iterative schemes such as simple iteration modified newton iteration or full newton iteration we employ a unifying approach for the construction of stochastic b series which is valid both for it o and stratonovich stochastic differential equations sdes and applicable both for weak and strong convergence to analyze the order of the iterated runge kutta method moreover the analytical techniques applied in this paper can be of use in many other similar contexts

symplectic integration of hamiltonian systems with additive noise hamiltonian systems with additive noise possess the property of preserving symplectic structure numerical methods with the same property are constructed for such systems special attention is paid to systems with separable hamiltonians and to second order differential equations with additive noise some numerical tests are presented

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

the kinetic scheme for the full burnett equations the present paper concerns two aspects for the burnett equations first we are going to theoretically show the consistency between the traditional chapman enskog expansion and the successive approximation for the bgk equation up to the super burnett order second we will design a numerical scheme to efficiently solve the burnett equations the current approach is an improvement of the bgk burnett scheme k xu phys fluids 15 2003 2077 where the locally constant collision frequency is considered based on the burnett distribution function a high order time accurate numerical flux is derived by different approaches the resulting scheme is tested in the problem of normal shock wave and that of force driven poiseuille flow

analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit we present a mathematical analysis of the asymptotic preserving scheme proposed in m lemou and l mieussens siam j sci comput 31 2008 pp 334 368 for linear transport equations in kinetic and diffusive regimes we prove that the scheme is uniformly stable and accurate with respect to the mean free path of the particles this property is satisfied under an explicitly given cfl condition this condition tends to a parabolic cfl condition for small mean free paths and is close to a convection cfl condition for large mean free paths our analysis is based on very simple energy estimates

coupling boltzmann and navier stokes equations by friction the aim of this paper is to introduce and validate a coupled navier stokes boltzman approach for the calculation of hypersonic rarefied flows around manoeuvering vehicles the proposed strategy uses locally a kinetic model in the boundary layer coupled through wall friction forces to a global navier stokes solver dif ferent numerical experiments illustrate the potentialities of the method

a stochastic immersed boundary method for fluid structure dynamics at microscopic length scales in modeling many biological systems it is important to take into account flexible structures which interact with a fluid at the length scale of cells and cell organelles thermal fluctuations of the aqueous environment become significant in this work it is shown how the immersed boundary method of c s peskin the immersed boundary method acta num 11 2002 1 39 for modeling flexible structures immersed in a fluid can be extended to include thermal fluctuations a stochastic numerical method is proposed which deals with stiffness in the system of equations by handling systematically the statistical contributions of the fastest dynamics of the fluid and immersed structures over long time steps an important feature of the numerical method is that time steps can be taken in which the degrees of freedom of the fluid are completely underresolved partially resolved or fully resolved while retaining a good level of accuracy error estimates in each of these regimes are given for the method a number of theoretical and numerical checks are furthermore performed to assess its physical fidelity for a conservative force the method is found to simulate particles with the correct boltzmann equilibrium statistics it is shown in three dimensions that the diffusion of immersed particles simulated with the method has the correct scaling in the physical parameters the method is also shown to reproduce a well known hydrodynamic effect of a brownian particle in which the velocity autocorrelation function exhibits an algebraic 3 2 decay for long times b j alder t e wainwright decay of the velocity autocorrelation function phys rev a 1 1 1970 18 21 a few preliminary results are presented for more complex systems which demonstrate some potential application areas of the method specifically we present simulations of osmotic effects of molecular dimers worm like chain polymer knots and a basic model of a molecular motor immersed in fluid subject to a hydrodynamic load force the theoretical analysis and numerical results show that the immersed boundary method with thermal fluctuations captures many important features of small length scale hydrodynamic systems and holds promise as an effective method for simulating biological phenomena on the cellular and subcellular length scales

combining word problems through rewriting in categories with products we give an algorithm solving combined word problems over non necessarily disjoint signatures based on rewriting of equivalence classes of terms the canonical rewriting system we introduce consists of few transparent rules and is obtained by applying knuth bendix completion procedure to presentations of pushouts among categories with products it applies to pairs of theories which are both constructible over their common reduct on which we do not make any special assumption

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

solving the boltzmann equation in n log 2 n in c mouhot and l pareschi fast algorithms for computing the boltzmann collision operator math comp to appear c mouhot and l pareschi c r math acad sci paris 339 2004 pp 71 76 fast deterministic algorithms based on spectral methods were derived for the boltzmann collision operator for a class of interactions including the hard spheres model in three dimensions these algorithms are implemented for the solution of the boltzmann equation in two and three dimensions first for homogeneous solutions then for general nonhomogeneous solutions the results are compared to explicit solutions when available and to monte carlo methods in particular the computational cost and accuracy are compared to those of monte carlo methods as well as to those of previous spectral methods finally for inhomogeneous solutions we take advantage of the great computational efficiency of the method to show an oscillation phenomenon of the entropy functional in the trend to equilibrium which was suggested in the work l desvillettes and c villani invent math 159 2005 pp 245 316

quasisymplectic integrators for stochastic differential equations two specialized algorithms for the numerical integration of the equations of motion of a brownian walker obeying detailed balance are introduced the algorithms become symplectic in the appropriate limits and reproduce the equilibrium distributions to some higher order in the integration time step comparisons with other existing integration schemes are carried out both for static and dynamical quantities

an efficient direct solver for rarefied gas flows with arbitrary statistics a new numerical methodology associated with a unified treatment is presented to solve the boltzmann bgk equation of gas dynamics for the classical and quantum gases described by the bose einstein and fermi dirac statistics utilizing a class of globally stiffly accurate implicit explicit runge kutta scheme for the temporal evolution associated with the discrete ordinate method for the quadratures in the momentum space and the weighted essentially non oscillatory method for the spatial discretization the proposed scheme is asymptotic preserving and imposes no non linear solver or requires the knowledge of fugacity and temperature to capture the flow structures in the hydrodynamic euler limit the proposed treatment overcomes the limitations found in the work by yang and muljadi 2011 33 due to the non linear nature of quantum relations and can be applied in studying the dynamics of a gas with internal degrees of freedom with correct values of the ratio of specific heat for the flow regimes for all knudsen numbers and energy wave lengths the present methodology is numerically validated with the unified treatment by the one dimensional shock tube problem and the two dimensional riemann problems for gases of arbitrary statistics descriptions of ideal quantum gases including rotational degrees of freedom have been successfully achieved under the proposed methodology

fast algorithms for computing the boltzmann collision operator the development of accurate and fast numerical schemes for the five fold boltzmann collision integral represents a challenging problem in scientific computing for a particular class of interactions including the so called hard spheres model in dimension three we are able to derive spectral methods that can be evaluated through fast algorithms these algorithms are based on a suitable representation and approximation of the collision operator explicit expressions for the errors in the schemes are given and spectral accuracy is proved parallelization properties and adaptivity of the algorithms are also discussed

stochastic symplectic and multi symplectic methods for nonlinear schrdinger equation with white noise dispersion we indicate that the nonlinear schrdinger equation with white noise dispersion possesses stochastic symplectic and multi symplectic structures based on these structures we propose the stochastic symplectic and multi symplectic methods which preserve the continuous and discrete charge conservation laws respectively moreover we show that the proposed methods are convergent with temporal order one in probability numerical experiments are presented to verify our theoretical results

coupling of the boltzmann and euler equations with automatic domain decomposition a criterion for automatic domain decomposition has been introduced for the coupling of the boltzmann and euler equations this criterion is obtained by using the grad s 13 moments expansion method the boltzmann and euler equations have been solved by the particle method this criterion has been further tested for the spatial homogeneous relaxation case and the coupling of the boltzmann and euler equations in two dimensional physical space some numerical results obtained from the particle code of the full boltzmann equation are compared with those of coupling of the boltzmann and euler equations

high order numerical methods for the space non homogeneous boltzmann equation in this paper we present accurate methods for the numerical solution of the boltzmann equation of rarefied gas the methods are based on a time splitting technique the transport is solved by a third order accurate in space positive and flux conservative pfc method the collision step is treated by a fourier approximation of the collision integral which guarantees spectral accuracy in velocity coupled with several high order integrators in time strang splitting is used to achieve second order accuracy in space and time several numerical tests illustrate the properties of the methods

a recursive sparse grid collocation method for differential equations with white noise we consider a sparse grid collocation method in conjunction with a time discretiza tion of the differential equations for computing expectations of functionals of solutions to differential equations perturbed by time dependent white noise we first analyze the error of smolyak s sparse grid collocation used to evaluate expectations of functionals of solutions to stochastic differential equations discretized by the euler scheme we show theoretically and numerically that this algo rithm can have satisfactory accuracy for small noise magnitude or small integration time however it does not converge either with decrease of the euler scheme s time step size or with increase of smolyak s sparse grid level subsequently we use this method as a building block for proposing a new algorithm by combining sparse grid collocation with a recursive procedure this approach allows us to numerically integrate linear stochastic partial differential equations over longer times which is illustrated in numerical tests on a stochastic advection diffusion equation

on the construction of kinetic schemes a general method for the construction of kinetic schemes of evolutionary equations is illustrated with the simple example of the linear advection equation where the role of the collision effect is clarified theoretically and numerically the application to the compressible euler equation and navier stokes equation is explained the theoretical backgrounds of the existing kinetic schemes such as the pullin scheme for the euler equation and chou baganoff scheme and modified prendergast xu scheme for the navier stokes equation are unified and their intrinsic errors are revealed

a multistage wiener chaos expansion method for stochastic advection diffusion reaction equations using wiener chaos expansion wce we develop numerical algorithms for solving second order linear parabolic stochastic partial differential equations spdes we propose a deterministic wce based algorithm for computing moments of the spde solutions without any use of the monte carlo technique we also compare the proposed deterministic algorithm with two other numerical methods based on the monte carlo technique and demonstrate that the new method is more efficient for highly accurate solutions numerical tests verify that the scheme is of mean square order o frac delta n 2 sqrt n 1 for diffusion and for diffusion reaction spdes with constant or variable coefficients where delta is the time step and n is the wiener chaos order

a multidimensional unified gas kinetic scheme for radiative transfer equations on unstructured mesh abstract in order to extend the unified gas kinetic scheme ugks to solve radiative transfer equations in a complex geometry a multidimensional asymptotic preserving implicit method on unstructured mesh is constructed in this paper with an implicit formulation the cfl condition for the determination of the time step in ugks can be much relaxed and a large time step is used in simulations differently from previous direction by direction ugks on orthogonal structured mesh on unstructured mesh the interface flux transport takes into account multi dimensional effect where gradients of radiation intensity and material temperature in both normal and tangential directions of a cell interface are included in the flux evaluation the multiple scale nature makes the ugks be able to capture the solutions in both optically thin and thick regions seamlessly in the optically thick region the condition of cell size being less than photon s mean free path is fully removed and the ugks recovers a solver for diffusion equation in such a limit on unstructured mesh for a distorted quadrilateral mesh the ugks goes to a nine point scheme for the diffusion equation and it naturally reduces to the standard five point scheme for a orthogonal quadrilateral mesh numerical computations covering a wide range of transport regimes on unstructured and distorted quadrilateral meshes will be presented to validate the current approach

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