a second order kinetic scheme for gas dynamics on arbitrary grids cartesian meshes for domains with complicated boundaries give rise to cut cells with arbitrarily small volumes explicit integration schemes over such meshes have a time step restriction proportional to the smallest cell volume we present an implementation of the kinetic scheme for gas dynamics by perthame b perthame boltzmann type schemes for gas dynamics and the entropy property siam j num anal 27 1990 1405 1421 on arbitrary cartesian meshes the formulation allows a time step based on the underlying regular cell size and retains l 1 stability positivity and second order convergence numerical convergence studies on arbitrary grids are presented

a hill series approach to wavefunction nodes the hill series method is shown to have several advantages for treating the schrodinger equation when the wavefunction obeys the dirichlet boundary condition r 0 a novel combination of the virial theorem the hill series method and a finite difference method is used to find the node positions of the wavefunction for several excited states of perturbed oscillator and coulombic systems and to find critical radii for the hydrogen atom

on the schrodinger equation for the gaussian potential a exp λr2 the energy eigenvalues of the attractive radial gaussian potential for various eigenstates are very accurately determined within the framework of the hypervirial pade scheme

an asymptotic preserving monte carlo method for the boltzmann equation abstract in this work we propose an asymptotic preserving monte carlo method for the boltzmann equation that is more efficient than the currently available monte carlo methods in the fluid dynamic regime this method is based on the successive penalty method 39 which is an improved bgk penalization method originally proposed by filbet and jin 16 here we introduce the monte carlo implementation of the method which despite its lower order accuracy is very efficient in higher dimensions or simulating some complicated chemical processes this method allows the time step independent of the mean free time which is prohibitively small in the fluid dynamic regime we study some basic properties of this method and compare it with some other asymptotic preserving monte carlo methods in terms of numerical performance in different regimes from rarefied to fluid dynamic regimes and their computational efficiency

an implicit monte carlo method for rarefied gas dynamics for the space homogeneous boltzmann equation we formulate a hybrid monte carlo method that is robust in the fluid dynamic limit this method is based on an analytic representation of the solution over a single time step and involves implicit time differencing derived from a suitable power series expansion of the solution a generalized wild expansion a class of implicit yet explicitly implementable numerical schemes is obtained by substituting a maxwellian distribution in place of the high order terms in the expansion the numerical solution is represented as a convex combination of a non equilibrium particle distribution and a maxwellian the hybrid distribution is then evolved by monte carlo using the implicit formulation for the time evolution computational simulations of spatially homogeneous problems by our method are presented here for the kac model and for the variable hard sphere model including maxwell molecules comparison to exact solutions and to direct simulation monte carlo dsmc computations shows the robustness and the efficiency of the new method

numerical comparison between the boltzmann and es bgk models for rarefied gases abstract rarefied gas flows obey the boltzmann equation but numerical simulations of this equation are not always possible so that simpler models have been introduced the es bgk equation is one of these models it gives the correct transport coefficients for the navier stokes approximation so that boltzmann or es bgk simulations are expected to give the same results for dense gases but in the case of a rarefied flow complete numerical comparisons are needed in this paper we present numerical comparisons between the two models in transitional regimes where the es bgk model is expected to be useful for reentry flows around a compression ramp and a plate we also emphasize that the es bgk model gives flow predictions closer to the boltzmann result than the simpler bgk model

an asymptotic preserving monte carlo method for the multispecies boltzmann equation an asymptotic preserving ap scheme is efficient in solving multiscale kinetic equations with a wide range of the knudsen number in this paper we generalize the asymptotic preserving monte carlo method ap dsmc developed in 25 to the multispecies boltzmann equation this method is based on the successive penalty method 26 originated from the bgk penalization based ap scheme developed in 7 for the multispecies boltzmann equation the penalizing maxwellian should use the unified maxwellian as suggested in 12 we give the details of ap dsmc for multispecies boltzmann equation show its ap property and verify through several numerical examples that the scheme can allow time step much larger than the mean free time thus making it much more efficient for flows with possibly small knudsen numbers than the classical dsmc

high order entropy based closures for linear transport in slab geometry ii a computational study of the optimization problem we present a numerical algorithm to implement entropy based m n moment models in the context of a simple linear kinetic equation for particles moving through a material slab the closure for these models as is the case for all entropy based models is derived through the solution of a constrained convex optimization problem the algorithm has two components the first component is a discretization of the moment equations which preserves the set of realizable moments thereby ensuring that the optimization problem has a solution in exact arithmetic the discretization is a second order kinetic scheme which uses muscl type limiting in space and a strong stability preserving runge kutta time integrator the second component of the algorithm is a newton based solver for the dual optimization problem which uses an adaptive quadrature to evaluate integrals in the dual objective and its derivatives the accuracy of the numerical solution to the dual problem plays a key role in the time step restric

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

runge kutta methods for hyperbolic conservation laws with stiff relaxation terms underresolved numerical schemes for hyperbolic conservation laws with stiff relaxation terms may generate unphysical spurious numerical results or reduce to lower order if the small relaxation time is not temporally well resolved we design a second order runge kutta type splitting method that possesses the discrete analogue of the continuous asymptotic limit which thus is able to capture the correct physical behaviors with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved

on the time relaxed monte carlo computations for the lid driven micro cavity flow abstract in the present study the first second and third orders of the trmc scheme trmc1 trmc2 and trmc3 schemes are employed to numerically investigate the lid driven micro cavity flow with different lid velocities and knudsen numbers the simulations cover flow regimes from early slip to early transition regimes 0 005 k n 0 1 higher order terms in the wild sum expansion are considered to obtain higher order collisions the results are compared with those from the standard dsmc method comparisons show that among the studied schemes the results obtained from the trmc3 scheme have excellent agreement with the ones from the dsmc method on the other hand the results of trmc1 and trmc2 schemes show deviations compared to those from the dsmc method the deviations are more pronounced for the velocity distributions moreover the present investigation illustrates that as the knudsen number increases the accuracy of lower orders of the trmc scheme improves it is observed that truncating the wild sum expansion up to the third order approximation of the trmc scheme may be a proper alternative method for the dsmc method in simulating the lid driven micro cavity problem for knudsen numbers 0 005 k n 0 1 with reasonable accuracy and simplicity in mathematics

an h adaptive mesh method for boltzmann bgk hydrodynamics coupling we introduce a coupled method for hydrodynamic and kinetic equations on 2 dimensional h adaptive meshes we adopt the euler equations with a fast kinetic solver in the region near thermodynamical equilibrium while use the boltzmann bgk equation in kinetic regions where fluids are far from equilibrium a buffer zone is created around the kinetic regions on which a gradually varying numerical flux is adopted based on the property of a continuously discretized cut off function which describes how the flux varies the coupling will be conservative in order for the conservative 2 dimensional specularly reflective boundary condition to be implemented conveniently the discrete maxwellian is approximated by a high order continuous formula with improved accuracy on a disc instead of on a square domain the h adaptive method can work smoothly with a time split numerical scheme through h adaptation the cell number is greatly reduced this method is particularly suitable for problems with hydrodynamics breakdown on only a small part of the whole domain so that the total efficiency of the algorithm can be greatly improved three numerical examples are presented to validate the proposed method and demonstrate its efficiency

numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation hyperbolic systems of conservation laws often have diffusive relaxation terms that lead to a small relaxation limit governed by reduced systems of a parabolic or hyperbolic type in such systems the understanding of basic wave pattern is difficult to achieve and standard high resolution methods fail to describe the right asymptotic behavior unless the small relaxation rate is numerically resolved we develop high resolution underresolved numerical schemes that possess the discrete analogue of the continuous asymptotic limit which are thus able to approximate the equilibrium system with high order accuracy even if the limiting equations may change type

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

construction of bgk models with a family of kinetic entropies for a given system of conservation laws we introduce a general framework for kinetic bgk models we assume to be given a system of hyperbolic conservation laws with a family of lax entropies and we characterize the bgk models that lead to this system in the hydrodynamic limit and that are compatible with the whole family of entropies this is obtained by a new characterization of maxwellians as entropy minimizers that can take into account the simultaneous minimization problems corresponding to the family of entropies we deduce a general procedure to construct such bgk models and we show how classical examples enter the framework we apply our theory to isentropic gas dynamics and full gas dynamics and in both cases we obtain new bgk models satisfying all entropy inequalities

moment methods for the semiconductor boltzmann equation on bounded position domains galerkin methods for the semiconductor boltzmann equation based on moment expansions for a discretization in the velocity direction are studied for the moment equations boundary conditions are proposed which are analogues to inflow and respectively reflecting boundary conditions for the boltzmann equation stability and an error estimate are proved for an expansion in terms of hermite polynomials finally an adaptive numerical implementation is introduced and results of numerical experiments are presented

on the hill determinant method explicit examples are given of the false eigenvalues of the hill determinant method the existence of which has been suspected by workers using analytic methods a simple criterion is discovered for distinguishing between false and physical eigenvalues

entropy satisfying flux vector splittings and kinetic bgk models we establish forward and backward relations between entropy satisfying bgk relaxation models such as those introduced previously by the author and the first order flux vector splitting numerical methods for general systems of conservation laws classically to a kinetic bgk model that is compatible with some family of entropies we can associate an entropy flux vector splitting we prove that the converse is true any entropy flux vector splitting can be interpreted by a kinetic model and we obtain an explicit characterization of entropy satisfying flux vector splitting schemes we deduce a new proof of discrete entropy inequalities under a sharp cfl condition that generalizes the monotonicity criterion in the scalar case in particular this gives a stability condition for numerical kinetic methods with noncompact velocity support a new interpretation of general kinetic schemes is also provided via approximate riemann solvers we deduce the construction of finite velocity relaxation systems for gas dynamics and obtain a hllc scheme for which we are able to prove positiveness of density and internal energy and discrete entropy inequalities

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

shooting methods for the schrodinger equation a simple stable shooting method for the schrodinger equation is described and is shown to work with various integrators including the numerov one quantum mechanical techniques are extended to apply to problems for which e is not a traditional eigenvalue and are shown to permit the extraction of useful information even from an unstable shooting method

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 local steiner type formula for general closed sets and applications we introduce support curvature measures of an arbitrary closed set a in d and establish a local steiner type formula for the localized parallel volume of a we derive some of the basic properties of these support measures and explore how they are related to the curvature measures available in the literature then we use the support measures in analysing contact distributions of stationary random closed sets with a particular emphasis on the boolean model with general compact particles

direct expectation value calculations a new application of the hellmann feynman theorem is shown to allow accurate calculation of expectation values along with energies for the schrodinger equation for variable boundary position and boundary conditions power series and finite difference versions of the method are developed and applied to perturbed oscillator and perturbed hydrogen atom problems

a high order monte carlo algorithm for the direct simulation of boltzmann equation a high order algorithm of the direct simulation monte carlo dsmc method h dsmc has been developed to simulate rarefied flow regimes the mth order taylor series expansion has been employed to obtain a more generalized form of the time discretization for the collision part of boltzmann equation in the purposed algorithm the higher order collision terms are introduced as well as higher order terms in the time step of the probabilistic coefficients these newly implemented higher order terms improve the accuracy and efficiency of the solution and enhance the convergence rate quite significantly comparison between results of the classic dsmc method and the h dsmc method shows the promising performance of the introduced technique

maps of convex sets and invariant regions for finite difference systems of conservation laws general results about maps of convex sets in r n are proved we outline their extensions to an infinite dimensional context such extensions have applications in nonlinear analysis such as in the study of the invariance of convex sets under nonlinear maps here we explore applications only in the finite dimensional context more specifically we apply the general results to the problem of finding sufficient conditions for a region of the state space to be globally or locally invariant under finite difference schemes applied to systems of conservation laws in several space variables in particular we establish a final characterization of the invariant regions under the lax friedrichs scheme and also give sufficient conditions for the local invariance further we give sufficient conditions for the global and local invariance of regions under flux splitting finite difference schemes an example of the multi dimensional euler equations for non isentropic gas dynamics is discussed

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

fluid solver independent hybrid methods for multiscale kinetic equations in some recent works g dimarco and l pareschi comm math sci 1 2006 pp 155 177 multiscale model simul 6 2008 pp 1169 1197 we developed a general framework for the construction of hybrid algorithms which are able to face efficiently the multiscale nature of some hyperbolic and kinetic problems here in contrast to previous methods we construct a method form fitting to any type of finite volume or finite difference scheme for the reduced equilibrium system thanks to the coupling of monte carlo techniques for the solution of the kinetic equations with macroscopic methods for the limiting fluid equations we show how it is possible to solve multiscale fluid dynamic phenomena faster than using traditional deterministic stochastic methods for the full kinetic equations in addition due to the hybrid nature of the schemes the numerical solution is affected by fewer fluctuations than are standard monte carlo schemes applications to the boltzmann bgk equation are presented to show the performance of the new methods in comparison with classical approaches used in the simulation of kinetic equations

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

numerical schemes for hyperbolic conservation laws with stiff relaxation terms hyperbolic systems often have relaxation terms that give them a partially conservative form and that lead to a long time behavior governed by reduced systems that are parabolic in nature in this article it is shown by asymptotic analysis and numerical examples that semidiscrete high resolution methods for hyperbolic conservation laws fail to capture this asymptotic behavior unless the small relaxation rate is resolved by a fine spatial grid we introduce a modification of higher order godunov methods that possesses the correct asymptotic behavior allowing the use of coarse grids large cell peclet numbers the idea is to build into the numerical scheme the asymptotic balances that lead to this behavior numerical experiments on 2 2 systems verify our analysis

convergence of a weighted particle method for solving the boltzmann b g k equation we consider in this paper the numerical solution of the b g k model for the boltzmann equation the numerical method used here was introduced by mas gallic transport theory statist phys 16 1987 pp 885 887 using the results of perthame and pulvirenti arch rational mech anal 125 1993 pp 289 295 we first derive a uniqueness result for the b g k model and also regularity properties of the solution which are necessary for our analysis we then prove following mas gallic and poupaud transport theory statist phys 17 1988 pp 311 345 the convergence of the numerical approximation for the b g k model finally some numerical experiments are presented

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

kinetic derivation of a finite difference scheme for the incompressible navier stokes equation in the present paper the low mach number limit of kinetic equations is used to develop a discretization for the incompressible navier stokes equation the kinetic equation is discretized with a first and second order discretization in space the discretized equation is then considered in the low mach number limit using this limit a second order discretization for the convective part in the incompressible navier stokes equation is obtained numerical experiments are shown comparing different approaches

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

time relaxed monte carlo methods for the boltzmann equation a new family of monte carlo schemes is introduced for the numerical solution of the boltzmann equation of rarefied gas dynamics the schemes are inspired by the wild sum expansion of the solution of the boltzmann equation for maxwellian molecules and consist of a novel time discretization of the equation in particular high order terms in the expansion are replaced by the equilibrium maxwellian distribution the two main features of the schemes are high order accuracy in time and asymptotic preservation the first property allows to recover accurate solutions with time steps larger than those required by direct simulation monte carlo dsmc while the latter guarantees that for the vanishing knudsen number the numerical solution relaxes to the local maxwellian conservation of mass momentum and energy are preserved by the scheme numerical results on several space homogeneous problems show the improvement of the new schemes over standard dsmc applications to a one dimensional shock wave problem are also presented

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

hill determinants and 1 n theory a computationally efficient form of the hill determinant method is devised giving both e and x2 values and is used to test the accuracy of that method in the light of recent criticisms l 1 and n 1 asymptotic theories are used to check the numerical results

an algebraic approach to the exponential cosine screened coulomb potential an analytical formula is obtained for the energy eigenvalues of the exponential cosine screened coulomb potential by a method that is a combination of algebraic perturbation theory based on the lie algebra of the so 2 1 group and the scaling variational method the results are found to be in excellent agreement with pade approximation calculations

discontinuous galerkin method for navier stokes equations using kinetic flux vector splitting kinetic schemes for compressible flow of gases are constructed by exploiting the connection between boltzmann equation and the navier stokes equations this connection allows us to construct a flux splitting for the navier stokes equations based on the direction of molecular motion from which a numerical flux can be obtained the naive use of such a numerical flux function in a discontinuous galerkin dg discretization leads to an unstable scheme in the viscous dominated case stable schemes are constructed by adding additional terms either in a symmetric or non symmetric manner which are motivated by the dg schemes for elliptic equations the novelty of the present scheme is the use of kinetic fluxes to construct the stabilization terms in the symmetric case interior penalty terms have to be added for stability and the resulting schemes give optimal convergence rates in numerical experiments the non symmetric schemes lead to a cell energy entropy inequality but exhibit sub optimal convergence rates these properties are studied by applying the schemes to a scalar convection diffusion equation and the 1 d compressible navier stokes equations in the case of navier stokes equations entropy variables are used to construct stable schemes

outer minkowski content for some classes of closed sets we find conditions ensuring the existence of the outer minkowski content for d dimensional closed sets in mathbb r d in connection with regularity properties of their boundaries moreover we provide a class of sets including all sufficiently regular sets stable under finite unions for which the outer minkowski content exists it follows in particular that finite unions of sets with lipschitz boundary and a type of sets with positive reach belong to this class

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

explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems we design numerical schemes for nonlinear degenerate parabolic systems with possibly dominant convection these schemes are based on discrete bgk models where both characteristic velocities and the source term depend singularly on the relaxation parameter general stability conditions are derived and convergence is proved to the entropy solutions for scalar equations

two term spectral asymptotics for the dirichlet laplacian in a lipschitz domain we prove a two term weyl type asymptotic formula for sums of eigenvalues of the dirichlet laplacian in a bounded open set with lipschitz boundary moreover in the case of a convex domain we obtain a universal bound which correctly reproduces the first two terms in the asymptotics

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

the asymptotic diffusion limit of a linear discontinuous discretization of a two dimensional linear transport equation abstract consider a linear transport problem and let the mean free path and the absorption cross section be of size it is well known that one obtains a diffusion problem as tends to zero we discretize the transport problem on a fixed mesh independent of consider again the limit 0 and ask whether one obtains an accurate discretization of the continuous diffusion problem the answer is known to be affirmative for the linear discontinuous galerkin finite element discretization in one space dimension in this paper we ask whether the same result holds in two space dimensions we consider a linear discontinuous discretization based on rectangular meshes our main result is that the asymptotic limit of this discrete problem is not a discretization of the asymptotic limit of the continuous problem and thus that the discretization will be inaccurate in the asymptotic regime under consideration we also propose a modified scheme which has the correct asymptotic behavior for spatially periodic problems although not always for problems with boundaries we present numerical results confirming our formal asymptotic analysis

the exponential cosine screened potential the concept of forward nested multiplication is used to develop a unified power series hill determinant method for the calculation of schrodinger equation energy levels a numerical application is made to the exponential cosine screened potential

uniformly accurate schemes for hyperbolic systems with relaxation we develop high resolution shock capturing numerical schemes for hyperbolic systems with relaxation in such systems the relaxation time may vary from order 1 to much less than unity when the relaxation time is small the relaxation term becomes very strong and highly stiff and underresolved numerical schemes may produce spurious results usually one cannot decouple the problem into separate regimes and handle different regimes with different methods thus it is important to have a scheme that works uniformly with respect to the relaxation time using the broadwell model of the nonlinear boltzmann equation we develop a second order scheme that works effectively with a fixed spatial and temporal discretization for all ranges of the mean free path formal uniform consistency proof for a first order scheme and numerical convergence proof for the second order scheme are also presented we also make numerical comparisons of the new scheme with some other schemes this study is motivated by the reentry problem in hypersonic computations

kinetic approximation of a boundary value problem for conservation laws we design numerical schemes for systems of conservation laws with boundary conditions these schemes are based on relaxation approximations taking the form of discrete bgk models with kinetic boundary conditions the resulting schemes are riemann solver free and easily extendable to higher order in time or in space for scalar equations convergence is proved we show numerical examples including solutions of euler equations

extensible grids uniform sampling on a space filling curve summary we study the properties of points in generated by applying hilbert s space filling curve to uniformly distributed points in 0 1 for deterministic sampling we obtain a discrepancy of for d 2 for random stratified sampling and scrambled van der corput points we derive a mean squared error of for integration of lipschitz continuous integrands when d 3 these rates are the same as those obtained by sampling on d dimensional grids and they show a deterioration with increasing d the rate for lipschitz functions is however the best possible at that level of smoothness and is better than plain independent and identically distributed sampling unlike grids space filling curve sampling provides points at any desired sample size and the van der corput version is extensible in n we also introduce a class of piecewise lipschitz functions whose discontinuities are in rectifiable sets described via minkowski content although these functions may have infinite variation in the sense of hardy and krause they can be integrated with a mean squared error of it was previously known only that the rate was other space filling curves such as those due to sierpinski and peano also attain these rates whereas upper bounds for the lebesgue curve are somewhat worse as if the dimension were times as high

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

boltzmann type schemes for gas dynamics and the entropy property let h be a convex function a boltzmann equation is built following the b g k model for which h is an entropy in the fluid limit the compressible euler equations are obtained with a convex entropy naturally associated to h the motivation is to define numerical boltzmann schemes with finite speed of propagation corresponding to an equilibrium density function with bounded support the classical scheme obtained with h f f log f is based on the equilibrium density function exp v 2 2 which has infinite support

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