a hybrid method for hydrodynamic kinetic flow part ii coupling of hydrodynamic and kinetic models in this work we present a non stationary domain decomposition algorithm for multiscale hydrodynamic kinetic problems in which the knudsen number may span from equilibrium to highly rarefied regimes our approach is characterized by using the full boltzmann equation for the kinetic regime the compressible euler equations for equilibrium with a buffer zone in which the bgk es equation is used to represent the transition between fully kinetic to equilibrium flows in this fashion the boltzmann solver is used only when the collision integral is non stiff and the mean free path is of the same order as the mesh size needed to capture variations in macroscopic quantities thus in principle the same mesh size and time steps can be used in the whole computation moreover the time step is limited only by convective terms since the boltzmann solver is applied only in wholly kinetic regimes we use the reduced noise dsmc scheme we have proposed in part i of the present work this ensures a smooth exchange of information across the different domains with a natural way to construct interface numerical fluxes several tests comparing our hybrid scheme with full boltzmann dsmc computations show the good agreement between the two solutions on a wide range of knudsen numbers

perturbation theory for the eckart young mirsky theorem and the constrained total least squares problem abstract golub et al linear algebra appl 88 89 1987 317 327 j demmel siam j numer anal 24 1987 199 206 generalized the eckart young mirsky eym theorem which solves the problem of approximating a matrix by one of lower rank with only a specific rectangular subset of the matrix allowed to be changed based on their results this paper presents perturbation analysis for the eym theorem and the constrained total least squares problem ctls

fluid dynamic limits of kinetic equations i formal derivations the connection between kinetic theory and the macroscopic equations of fluid dynamics is described in particular our results concerning the incompressible navier stokes equations are based on a formal derivation in which limiting moments are carefully balanced rather than on a classical expansion such as those of hilbert or chapman enskog the moment formalism shows that the limit leading to the incompressible navier stokes equations like that leading to the compressible euler equations is a natural one in kinetic theory and is contrasted with the systematics leading to the compressible navier stokes equations some indications of the validity of these limits are given more specifically the connection between the diperna lions renormalized solution of the classical boltzmann equation and the leray solution of the navier stokes equations is discussed

numerical normal forms for codim 2 bifurcations of fixed points with at most two critical eigenvalues in this paper we derive explicit formulas for the normal form coefficients to verify the nondegeneracy of eight codimension two bifurcations of fixed points with one or two critical eigenvalues these include all strong resonances as well as the degenerate flip and neimark sacker bifurcations applying our results to n dimensional maps one avoids the computation of the center manifold but one can directly evaluate the critical normal form coefficients in the original basis the formulas remain valid also for n 2 and allow one to avoid the transformation of the linear part of the map into jordan form the developed techniques are tested on two examples 1 a three dimensional map appearing in adaptive control 2 a periodically forced epidemic model we compute numerically the critical normal form coefficients for several codim 2 bifurcations occurring in these models to compute the required derivatives of the poincare map for the epidemic model the automatic differentiation package adol c is used

the smallest perturbation of a submatrix which lowers the rank and constrained total least squares problems let t left begin array 20 c a b c d end array right be a partitioned rectangular matrix we exhibit perturbations of d of smallest norm which lower the rank of t we apply this result to generalize the total least squares problem sx b when only a rectangular subset of the data s b is considered uncertain

global characterization of the normal form for a vector field near a closed orbit on deduit une generalisation naturelle de la caracterisation globale de elphick pour des champs de vecteurs pres d une orbite fermee

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

a new asymptotic preserving scheme based on micro macro formulation for linear kinetic equations in the diffusion limit we propose a new numerical scheme for linear transport equations it is based on a decomposition of the distribution function into equilibrium and nonequilibrium parts we also use a projection technique that allows us to reformulate the kinetic equation into a coupled system of an evolution equation for the macroscopic density and a kinetic equation for the nonequilibrium part by using a suitable time semi implicit discretization our scheme is able to accurately approximate the solution in both kinetic and diffusion regimes it is asymptotic preserving in the following sense when the mean free path of the particles is small our scheme is asymptotically equivalent to a standard numerical scheme for the limit diffusion model a uniform stability property is proved for the simple telegraph model various boundary conditions are studied our method is validated in one dimensional cases by several numerical tests and comparisons with previous asymptotic preserving schemes

a hierarchy of hybrid numerical methods for multiscale kinetic equations in this paper we construct a hierarchy of hybrid numerical methods for multi scale kinetic equations based on moment realizability matrices a concept introduced by levermore morokoff and nadiga in phys fluids 10 1998 pp 3214 3226 following such a criterion one can consider a hybrid scheme where the hydrodynamic part is given either by the compressible euler or navier stokes equations or even with more general models such as the burnett or super burnett systems

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

uniformly accurate diffusive relaxation schemes for multiscale transport equations many transport equations such as the neutron transport radiative transfer and transport equations for waves in random media have a diffusive scaling that leads to the diffusion equations in many physical applications the scaling parameter mean free path may differ in several orders of magnitude from the rarefied regimes to the hydrodynamic diffusive regimes within one problem and it is desirable to develop a class of robust numerical schemes that can work uniformly with respect to this relaxation parameter in an earlier work jin pareschi and toscani siam j numer anal 35 1998 pp 2405 2439 we handled this numerical problem for discrete velocity kinetic models by reformulating the system into a form commonly used for a relaxation scheme for conservation laws jin and xin comm pure appl math 48 1995 pp 235 277 such a reformulation allows us to use the splitting technique for relaxation schemes to design a class of implicit yet explicitly implementable schemes that work with high resolution uniformly with respect to the relaxation parameter in this paper we show that such a numerical technique can be applied to a large class of transport equations with continuous velocities when one uses the even and odd parities of the transport equation

numerical periodic normalization for codim 1 bifurcations of limit cycles explicit computational formulas for the coefficients of the periodic normal forms for all codim 1 bifurcations of limit cycles in generic autonomous odes are derived they include second order coefficients for the fold limit point bifurcation as well as third order coefficients for the flip period doubling and neimark sacker torus bifurcations the formulas are independent of the dimension of the phase space and involve solutions of certain boundary value problems on the interval 0 t where t is the period of the critical cycle as well as multilinear functions from the taylor expansion of the right hand sides near the cycle the formulas allow us to distinguish between sub and supercritical bifurcations in agreement with earlier asymptotic expansions of the bifurcating solutions our formulation makes it possible to use robust numerical boundary value algorithms based on orthogonal collocation rather than shooting techniques which greatly expands its applicability the actual implementation is described in detail we include three numerical examples in which codim 2 singularities are detected along branches of codim 1 bifurcations of limit cycles as zeros of the periodic normal form coefficients

an asymptotic preserving automatic domain decomposition method for the vlasov poisson bgk system with applications to plasmas in this work we present an efficient strategy to deal with plasma physics simulations in which localized departures from thermodynamical equilibrium are present the method is based on the introduction of intermediate regions which allows smooth transitions between kinetic and fluid zones in this paper we extend domain decomposition techniques obtained through dynamic coupling and buffer zones to the study of plasmas and moreover we combine them with asymptotic preserving and asymptotically accurate strategies for the time integration we use a hybrid scheme in which both kinetic and fluid descriptions are considered and coupled together while the kinetic model is solved by asymptotic preserving and accurate methods in order to guarantee high efficiency and accuracy in all regimes the numerical scheme is validated and its performances are analyzed by numerical simulations

a multiscale kinetic fluid solver with dynamic localization of kinetic effects this paper collects the efforts done in our previous works 7 9 10 to build a robust multiscale kinetic fluid solver our scope is to efficiently solve fluid dynamic problems which present non equilibrium localized regions that can move merge appear or disappear in time the main ingredients of the present work are the followings ones a fluid model is solved in the whole domain together with a localized kinetic upscaling term that corrects the fluid model wherever it is necessary this multiscale description of the flow is obtained by using a micro macro decomposition of the distribution function 9 the dynamic transition between fluid and kinetic descriptions is obtained by using a time and space dependent transition function to efficiently define the breakdown conditions of fluid models we propose a new criterion based on the distribution function itself several numerical examples are presented to validate the method and measure its computational efficiency

high order imex spectral schemes for computing the dynamics of systems of nonlinear schrodinger gross pitaevskii equations the aim of this paper is to build and validate some explicit high order schemes both in space and time for simulating the dynamics of systems of nonlinear schrodinger gross pitaevskii equations the method is based on the combination of high order implicit explicit imex schemes in time and fourier pseudo spectral approximations in space the resulting imexsp schemes are highly accurate efficient and easy to implement they are also robust when used in conjunction with an adaptive time stepping strategy and appear as an interesting alternative to time splitting pseudo spectral tssp schemes finally a complete numerical study is developed to investigate the properties of the imexsp schemes in comparison with tssp schemes for one and two components systems of gross pitaevskii equations

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

asymptotic preserving particle in cell method for the vlasov poisson system near quasineutrality this paper deals with the numerical resolution of the vlasov poisson system in a nearly quasineutral regime by particle in cell pic methods in this regime classical pic methods are subject to stability constraints on the time and space steps related to the small debye length and large plasma frequency here we propose an asymptotic preserving pic scheme which is not subjected to these limitations additionally when the plasma period and debye length are small compared to the time and space steps this method provides a consistent pic discretization of the quasineutral vlasov equation we perform several one dimensional numerical experiments which provide a solid validation of the method and its underlying concepts and compare the method with classical pic and direct implicit methods

switching to nonhyperbolic cycles from codim 2 bifurcations of equilibria in odes the paper provides full algorithmic details on switching to the continuation of all possible codim 1 cycle bifurcations from generic codim 2 equilibrium bifurcation points in n dimensional odes we discuss the implementation and the performance of the algorithm in several examples including an extended lorenz 84 model and a laser system

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

analysis of bifurcations of limit cycles with lyapunov exponents and numerical normal forms in this paper we focus on the combination of normal form and lyapunov exponent computations in the numerical study of the three codim 2 bifurcations of limit cycles with dimension of the center manifold equal to 4 or to 5 in generic autonomous odes the normal form formulas are independent of the dimension of the phase space and involve solutions of certain linear boundary value problems the formulas allow one to distinguish between the complicated bifurcation scenarios which can happen near these codim 2 bifurcations where 3 tori and 4 tori can be present we apply our techniques to the study of a known laser model a novel model from population biology and a model of mechanical vibrations these models exhibit limit point neimark sacker period doubling neimark sacker and double neimark sacker bifurcations lyapunov exponents are computed to numerically confirm the results of the normal form analysis in particular with respect to the existence of stable invariant tori of various dimensions conversely the normal forms are essential to understand the significance of the lyapunov exponents

on perturbations of some constrained subspaces perturbation bounds of subspaces such as eigen spaces singular subspaces and canonical subspaces have been extensively studied in the literature in this paper we study perturbations of some constrained subspaces of 1x2 2x1 and 2x2 block matrices in which only one of the sub matrices can be changed such problems rise from the least squares total least squares problem the constrained least squares problem and the constrained total least squares problem

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

convergence of a discrete velocity model for the boltzmann bgk equation abstract we prove the convergence of a conservative and entropic discrete velocity model for the bathnagar gross krook bgk equation in this model the approximation of the maxwellian is based on a discrete entropy minimization principle the main difficulty due to its implicit definition is to prove that this approximation is consistent we also demonstrate the existence and uniqueness of a solution to the discrete velocity model by using a fixed point theorem finally the model is written in a continuous equation form and we prove the convergence of its solution toward a solution of the bgk equation

a generalization of the eckart young mirsky matrix approximation theorem abstract the eckart young mirsky theorem solves the problem of approximating a matrix by one of lower rank however the approximation generally differs from the original in all its elements in this paper it is shown how to obtain a best approximation of lower rank in which a specified set of columns of the matrix remains fixed the paper concludes with some applications of the generalization

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

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

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

analysis of the generalized total least squares problemax b when some columns of a are free of error this paper studies the algebraic properties and perturbation theory of the generalized total least squares problem gtls ax b in whicha a 1 a 2 a 1 is free of error and the error contained in a 2 b is of the formec withc a given nonsingular matrix the problem was proposed by van huffel and vandewalle in 15 the solvability conditions formulas for the gtls solutions their residuals and the minimum norm correction matrices are obtained and a perturbation theory for the gtls problem is given

existence uniqueness and positivity of solutions for bgk models for mixtures abstract we consider kinetic models for a multi component gas mixture without chemical reactions in the literature one can find two types of bgk models in order to describe gas mixtures one type has a sum of bgk type interaction terms in the relaxation operator for example the model described by klingenberg pirner and puppo 20 which contains well known models of physicists and engineers for example hamel 16 and gross and krook 15 as special cases the other type contains only one collision term on the right hand side for example the well known model of andries aoki and perthame 1 for each of these two models 20 and 1 we prove existence uniqueness and positivity of solutions in the first part of the paper in the second part we use the first model 20 in order to determine an unknown function in the energy exchange of the macroscopic equations for gas mixtures described by dellacherie 11

relaxation schemes for nonlinear kinetic equations a class of numerical schemes for nonlinear kinetic equations of boltzmann type is described following wild s approach the solution is represented as a power series with parameter depending exponentially on the knudsen number this permits us to derive accurate and stable time discretizations for all ranges of the mean free path these schemes preserve the main physical properties positivity conservation of mass momentum and energy moreover for some particular models the entropy property is also shown to hold

an adaptive domain decomposition procedure for boltzmann and euler equations in this paper we present a domain decomposition approach for the coupling of boltzmann and euler equations particle methods are used for both equations this leads to a simple implementation of the coupling procedure and to natural interface conditions between the two domains adaptive time and space discretizations and a direct coupling procedure lead to considerable gains in cpu time compared to a solution of the full boltzmann equation several test cases involving a large range of knudsen numbers are numerically investigated

a numerical method for kinetic semiconductor equations in the drift diffusion limit an asymptotic induced scheme for kinetic semiconductor equations with the diffusion scaling is developed the scheme is based on the asymptotic analysis of the kinetic semiconductor equation it works uniformly for all ranges of mean free paths the velocity discretization is done using quadrature points equivalent to a moment expansion method numerical results for different physical situations are presented

a moving interface method for dynamic kinetic fluid coupling this paper is a continuation of earlier work p degond s jin l mieussens a smooth transition between kinetic and hydrodynamic equations journal of computational physics 209 2005 665 694 in which we presented an automatic domain decomposition method for the solution of gas dynamics problems which require a localized resolution of the kinetic scale the basic idea is to couple the macroscopic hydrodynamics model and the microscopic kinetic model through a buffer zone in which both equations are solved discontinuities or sharp gradients of the solution are responsible for locally strong departures to local equilibrium which require the resolution of the kinetic model the buffer zone is drawn around the kinetic region by introducing a cut off function which takes values between zero and one and which is identically zero in the fluid zone and one in the kinetic zone in the present paper we specifically consider the possibility of moving the kinetic region or creating new kinetic regions by evolving the cut off function with respect to time we present algorithms which perform this task by taking into account indicators which characterize the non equilibrium state of the gas the method is shown to be highly flexible as it relies on the time evolution of the buffer cut off function rather than on the geometric definition of a moving interface which requires remeshing by contrast to many previous methods numerical examples are presented which validate the method and demonstrate its performances

numerical methods for two parameter local bifurcation analysis of maps we discuss new and improved algorithms for the bifurcation analysis of fixed points and periodic orbits cycles of maps and their implementation in matcont a matlab toolbox for continuation and bifurcation analysis of dynamical systems this includes the numerical continuation of fixed points of iterates of the map with one control parameter detecting and locating their bifurcation points i e limit point period doubling and neimark sacker and their continuation in two control parameters as well as detection and location of all codimension 2 bifurcation points on the corresponding curves for all bifurcations of codim 1 and 2 the critical normal form coefficients are computed both numerically with finite directional differences and using symbolic derivatives of the original map using a parameter dependent center manifold reduction explicit asymptotics are derived for bifurcation curves of double and quadruple period cycles rooted at codim 2 points of cycles with arbitrary period these asymptotics are implemented into the software and allow one to switch at codim 2 points to the continuation of the double and quadruple period bifurcations we provide two examples illustrating the developed techniques a generalized henon map and a juvenile adult competition model from mathematical biology

algebraic relations between the total least squares and least squares problems with more than one solution this paper completes our previous discussion on the total least squares tls and the least squares ls problems for the linear systemax b which may contain more than one solution 12 13 generalizes the work of golub and van loan 1 2 van huffel 8 van huffel and vandewalle 11 the tls problem is extended to the more general case the sets of the solutions and the squared residuals for the tls and ls problems are compared the concept of the weighted squares residuals is extended and the difference between the tls and the ls approaches is derived the connection between the approximate subspaces and the perturbation theories are studied it is proved that under moderate conditions all the corresponding quantities for the solution sets of the tls and the modified ls problems are close to each other while the quantities for the solution set of the ls problem are close to the corresponding ones of a subset of that of the tls problem

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

numerical normalization techniques for all codim 2 bifurcations of equilibria in ode s explicit computational formulas for the coefficients of the normal forms for all codim 2 equilibrium bifurcations of equilibria in autonomous odes are derived they include second order coefficients for the bogdanov takens bifurcation third order coefficients for the cusp and fold hopf bifurcations and coefficients of the fifth order terms for the generalized hopf bautin and double hopf bifurcations the formulas are independent on the dimension of the phase space and involve only critical eigenvectors of the jacobian matrix of the right hand sides and its transpose as well as multilinear functions from the taylor expansion of the right hand sides at the critical equilibrium the normal form coefficients for the fold hopf bifurcation in the new lorenz model are computed using the derived formulas proving the existence of a nontrivial invariant set in the system

computing periodic orbits and their bifurcations with automatic differentiation this paper formulates several algorithms for the direct computation of periodic orbits as solutions of boundary value problems the algorithms emphasize the use of coarse meshes and high orders of accuracy convergence theorems are given in the limit of increasing order with a fixed mesh the algorithms are implemented with the use of matlab and adol c a software package for automatic differentiation automatic differentiation enables accurate computation of high order derivatives of functions without the truncation errors inherent in finite difference calculations we embed the algorithms in a continuation framework and extend them to compute saddle node bifurcations of periodic orbits directly we present data from numerical studies of four test problems making some comparisons with other methods for computing periodic orbits these results demonstrate that high order methods based upon automatic differentiation are capable of high precision with small meshes

a combined continuum dsmc technique for multiscale analysis of microfluidic filters a multiscale method that combines continuum fluid models with the direct simulation monte carlo dsmc method is presented the continuum regions are treated by stokes equations and a scattered point based finite cloud method is employed to solve the stokes equations the continuum and dsmc regions are combined by an overlapped schwarz alternating method with dirichlet dirichlet type boundary conditions a scattered point interpolation scheme is developed to interpolate the solution between subdomains the convergence characteristics of the multiscale approach are investigated in detail specifically the dependence of convergence on the overlap size the dsmc noise and the number of time steps employed in the dsmc algorithm are studied while the convergence depends weakly on the dsmc noise and the overlap size the number of dsmc time steps simulated in each coupling iteration should be selected so that the total time steps simulated until convergence of the coupled process is close to the time constant of the dsmc subsystem steady state analysis of microfluidic filters is studied in detail using the multiscale approach the multiscale approach is also applied for the simulation of a membrane with an array of microfluidic filters and a dual stage microfluidic device with an array of microfluidic filters for particle trapping and sorting

computation of periodic solution bifurcations in odes using bordered systems we consider numerical methods for the computation and continuation of the three generic secondary periodic solution bifurcations in autonomous odes namely the fold the period doubling or flip bifurcation and the torus or neimark sacker bifurcation in the fold and flip cases we append one scalar equation to the standard periodic bvp that defines the periodic solution in the torus case four scalar equations are appended evaluation of these scalar equations and their derivatives requires the solution of linear bvps whose sparsity structure after discretization is identical to that of the linearization of the periodic bvp therefore the calculations can be done using existing numerical linear algebra techniques such as those implemented in the software auto and colsys

a smooth transition model between kinetic and hydrodynamic equations this paper presents a model which provides a smooth transition between a kinetic and a hydrodynamic domain the idea is to use a buffer zone in which both hydrodynamics and kinetic equations will be solved the solution of the original kinetic equation will be recovered as the sum of the solutions of these two equations we use an artificial connecting function which makes the equation on each domain degenerate at the end of the buffer zone thus no boundary condition is needed at the transition point consequently this model avoids the delicate issue of finding the interface condition in a typical domain decomposition method that couples a kinetic equation with hydrodynamic equations a simple kinetic scheme is developed to discretize our model and numerical examples are used to validate the method

asymptotic preserving implicit explicit runge kutta methods for nonlinear kinetic equations we discuss implicit explicit imex runge kutta methods which are particularly adapted to stiff kinetic equations of boltzmann type we consider both the case of easy invertible collision operators and the challenging case of boltzmann collision operators we give sufficient conditions in order that such methods are asymptotic preserving and asymptotically accurate their monotonicity properties are also studied in the case of the boltzmann operator the methods are based on the introduction of a penalization technique for the collision integral this reformulation of the collision operator permits us to construct penalized imex schemes which work uniformly for a wide range of relaxation times avoiding the expensive implicit resolution of the collision operator finally we show some numerical results which confirm the theoretical analysis

a hybrid kinetic fluid model for solving the gas dynamics boltzmann bgk equation our purpose is to derive a hybrid model for particle systems which combines a kinetic description of the fast particles with a fluid description of the thermal ones in the present work fast particles will be described through a collisional kinetic equation of boltzmann bgk type while thermal particles will be modeled by means of a system of euler type equations then we construct a numerical scheme for this model this scheme satisfies exact conservation properties we validate the approach by presenting various numerical tests