linearly implicit imex runge kutta methods for a class of degenerate convection diffusion problems multispecies kinematic flow models with strongly degenerate diffusive corrections give rise to systems of nonlinear convection diffusion equations of arbitrary size applications of these systems include models of polydisperse sedimentation and multiclass traffic flow implicit explicit imex runge kutta rk methods are suitable for the solution of these convection diffusion problems since the stability restrictions coming from the explicitly treated convective part are much less severe than those that would be deduced from an explicit treatment of the diffusive term these schemes usually combine an explicit rk scheme for the time integration of the convective part with a diagonally implicit one for the diffusive part in r burger p mulet and l m villada siam j sci comput 35 2013 pp b751 b777 a scheme of this type is proposed where the nonlinear and nonsmooth systems of algebraic equations arising in the implicit treatment of the degenerate diffusive part are solved by smoothing o

hyperbolic conservation laws with relaxation the effect of relaxation is important in many physical situations it is present in the kinetic theory of gases elasticity with memory gas flow with thermo non equilibrium water waves etc the governing equations often take the form of hyperbolic conservation laws with lower order terms in this article we present and analyze a simple model of hyperbolic conservation laws with relaxation effects dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified stability criteria for diffusion waves expansion waves and traveling waves are found and justified nonlinearly time asymptotic expansion and the energy method are used in the analysis for dissipative waves the expansion is similar in spirit to the chapman enskog expansion in the kinetic theory for shock waves however a different approach is needed

a modified higher order godunov s scheme for stiff source conservative hydrodynamics we present an efficient second order accurate scheme to treat stiff source terms within the framework of higher order godunov s methods we employ duhamel s formula to devise a modified predictor step which accounts for the effects of stiff source terms on the conservative fluxes and recovers the correct isothermal behavior in the limit of an infinite cooling reaction rate source term effects on the conservative quantities are fully accounted for by means of a one step second order accurate semi implicit corrector scheme based on the deferred correction method of dutt et al we demonstrate the accurate stable and convergent results of the proposed method through a set of benchmark problems for a variety of stiffness conditions and source types

high resolution staggered mesh approach for nonlinear hyperbolic systems of conservation laws abstract a numerical technique is presented to approximate weak solutions of hyperbolic systems of conservation laws in one and two space dimensions when strong shocks are present in the exact solution other techniques rely on approximate riemann problem solvers to cope with this difficulty the high order approach here does not use approximate riemann solvers as a building block and is therefore considerably easier to implement compared to the usual methods moreover the approach here yields sharp nonoscillatory shocks sharp corners at the base of rarefaction waves and has high order accuracy in regions where the solution is smooth for the scalar one dimensional problem the theoretical results confirm the reliability of this approach

a high order kinetic flux vector splitting method for the reduced five equation model of compressible two fluid flows we present a high order kinetic flux vector splitting kfvs scheme for the numerical solution of a conservative interface capturing five equation model of compressible two fluid flows this model was initially introduced by wackers and koren 2004 21 the flow equations are the bulk equations combined with mass and energy equations for one of the two fluids the latter equation contains a source term in order to account for the energy exchange we numerically investigate both one and two dimensional flow models the proposed numerical scheme is based on the direct splitting of macroscopic flux functions of the system of equations in two space dimensions the scheme is derived in a usual dimensionally split manner the second order accuracy of the scheme is achieved by using muscl type initial reconstruction and runge kutta time stepping method for validation the results of our scheme are compared with those from the high resolution central scheme of nessyahu and tadmor 14 the accuracy efficiency and simplicity of the kfvs scheme demonstrate its potential for modeling two phase flows

numerical methods for solving moment equations in kinetic theory of neuronal network dynamics recently developed kinetic theory and related closures for neuronal network dynamics have been demonstrated to be a powerful theoretical framework for investigating coarse grained dynamical properties of neuronal networks the moment equations arising from the kinetic theory are a system of 1 1 dimensional nonlinear partial differential equations pde on a bounded domain with nonlinear boundary conditions the pdes themselves are self consistently specified by parameters which are functions of the boundary values of the solution the moment equations can be stiff in space and time numerical methods are presented here for efficiently and accurately solving these moment equations the essential ingredients in our numerical methods include i the system is discretized in time with an implicit euler method within a spectral deferred correction framework therefore the pdes of the kinetic theory are reduced to a sequence in time of boundary value problems bvps with nonlinear boundary conditions ii a set of auxiliary parameters is introduced to recast the original bvp with nonlinear boundary conditions as bvps with linear boundary conditions with additional algebraic constraints on the auxiliary parameters iii a careful combination of two newton s iterates for the nonlinear bvp with linear boundary condition interlaced with a newton s iterate for solving the associated algebraic constraints is constructed to achieve quadratic convergence for obtaining the solutions with self consistent parameters it is shown that a simple fixed point iteration can only achieve a linear convergence for the self consistent parameters the practicability and efficiency of our numerical methods for solving the moment equations of the kinetic theory are illustrated with numerical examples it is further demonstrated that the moment equations derived from the kinetic theory of neuronal network dynamics can very well capture the coarse grained dynamical properties of integrate and fire neuronal networks

second order boltzmann schemes for compressible euler equations in one and two space dimensions a class of second order numerical schemes for the compressible euler equations is described and their l 1 stability i e rho geqq 0 t geqq 0 is proved following van leer s approach the solution rho u sqrt t here is represented as piecewise linear functions the necessity of a slope limitation appears naturally in the derivation of the schemes but it can be less strict than the slope reconstructions usually used these schemes are written in terms of explicit flux splitting formula and are naturally multidimensional in space the upwinding is obtained through a very generalized notion of characteristics the kinetic one

combinatorial formulas for vassiliev invariants from chern simons gauge theory we analyze the perturbative series expansion of the vacuum expectation value of a wilson loop in chern simons gauge theory in the temporal gauge from the analysis emerges the notion of the kernel of a vassiliev invariant the kernel of a vassiliev invariant of order n is not a knot invariant since it depends on the regular knot projection chosen but it differs from a vassiliev invariant by terms that vanish on knots with n singular crossings we conjecture that vassiliev invariants can be reconstructed from their kernels we present the general form of the kernel of a vassiliev invariant and we describe the reconstruction of the full primitive vassiliev invariants at orders two three and four at orders two and three we recover known combinatorial expressions for these invariants at order four we present new combinatorial expressions for the two primitive vassiliev invariants present at this order

why the muscl hancock scheme is l 1 stable the finite volume methods are one of the most popular numerical procedure to approximate the weak solutions of hyperbolic systems of conservation laws they are developed in the framework of first order numerical schemes several approaches are proposed to increase the order of accuracy the van leer methods are interesting ways one of them namely the muscl hancock scheme is full time and space second order accuracy in the present work we exhibit relevant conditions to ensure the l1 stability of the method a cfl like condition is established and a suitable limitation procedure for the gradient reconstruction is developed in order to satisfy the stability criterion in addition we show that the conservative variables are not useful within the gradient reconstruction and the procedure is extended in the framework of the primitive variables numerical experiments are performed to show the interest and the robustness of the method

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

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

kinetic flux vector splitting schemes for the hyperbolic heat conduction a kinetic solver is developed for the initial and boundary value problems of the symmetric hyperbolic moment system this nonlinear system of equations is related to the heat conduction in solids at low temperatures the system consists of a conservation equation for the energy density e and a balance equation for the heat flux qi where e and qi are the four basic fields of the theory we use kinetic flux vector splitting kfvs scheme to solve these equations in one and two space dimensions the flux vectors of the equations are splitted on the basis of the local equilibrium distribution of phonons the resulting computational procedure is efficient and straightforward to implement the second order accuracy of the scheme is achieved by using muscl type reconstruction and min mod nonlinear limiters the solutions exhibit second order accuracy and satisfactory resolution of gradients with no spurious oscillations the scheme is extended to the two dimensional case in a usual dimensionally split manner in order to prescribe the boundary data we need the knowledge of the e and qi however in experiments only one of the quantities can be controlled at the boundary this problem is removed by using a continuity condition it turned out that after some short time energy and heat flux are related to each other according to rankine hugoniot jump relations to illustrate the performance of the kfvs scheme we perform several one and two dimensional test computations for the comparison of our results we use high order central schemes the present study demonstrates that this kinetic method is effective in handling such problems

numerical methods for hyperbolic conservation laws with stiff relaxation ii higher order godunov methods a higher order godunov method is presented for hyperbolic systems of conservation laws with stiff relaxing source terms the goal is to develop a godunov method that produces higher order accurate

a general approach to heisenberg categorification via wreath product algebras we associate a monoidal category mathcal h b defined in terms of planar diagrams to any graded frobenius superalgebra b this category acts naturally on modules over the wreath product algebras associated to b to b we also associate a quantum lattice heisenberg algebra mathfrak h b we show that provided b is not concentrated in degree zero the grothendieck group of mathcal h b is isomorphic as an algebra to mathfrak h b for specific choices of frobenius algebra b we recover existing results including those of khovanov and cautis licata we also prove that certain morphism spaces in the category mathcal h b contain generalizations of the degenerate affine hecke algebra specializing b this proves an open conjecture of cautis licata

automatic classification of protein structure by using gauss integrals we introduce a method of looking at analyzing and comparing protein structures the topology of a protein is captured by 30 numbers inspired by vassiliev knot invariants to illustrate the simplicity and power of this topological approach we construct a measure scaled gauss metric sgm of similarity of protein shapes under this metric protein chains naturally separate into fold clusters we use sgm to construct an automatic classification procedure for the cath2 4 database the method is very fast because it requires neither alignment of the chains nor any chain chain comparison it also has only one adjustable parameter we assign 95 51 of the chains into the proper c class a architecture t topology and h homologous superfamily fold find all new folds and detect no false geometric positives using the sgm we display a map of the space of folds projected onto two dimensions show the relative locations of the major structural classes and zoom into the space of proteins to show architecture topology and fold clusters the existence of a simple measure of a protein fold computed from the chain path will have a major impact on automatic fold classification

an asymptotic induced scheme for nonstationary transport equations in the diffusive limit an asymptotic induced scheme for nonstationary transport equations with the diffusion scaling is developed the scheme works uniformly for all ranges of mean free paths it is based on the asymptotic analysis of the diffusion limit of the transport equation a theoretical investigation of the behavior of the scheme in the diffusion limit is given and an approximation property is proven moreover numerical results for different physical situations are shown and the uniform convergence of the scheme is established numerically

global entropy solutions to exothermically reacting compressible euler equations abstract the global existence of entropy solutions is established for the compressible euler equations for one dimensional or plane wave flow of an ideal gas which undergoes a one step exothermic chemical reaction under arrhenius type kinetics we assume that the reaction rate is bounded away from zero and the total variation of the initial data is bounded by a parameter that grows arbitrarily large as the equation of state converges to that of an isothermal gas the heat released by the reaction causes the spatial total variation of the solution to increase however the increase in total variation is proved to be bounded in t 0 as a result of the uniform and exponential decay of the reactant to zero as t approaches infinity

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

exact maximum entropy closure of the hydrodynamical model for si semiconductors the 8 moment case an exact closure is obtained of the 8 moment model for semiconductors based on the maximum entropy principle in the case of silicon semiconductors the validity of the model is assessed and comparisons with an approximate closure are presented and discussed

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

central differencing based numerical schemes for hyperbolic conservation laws with relaxation terms many applications involve hyperbolic systems of conservation laws with source terms the numerical solution of such systems may be challenging especially when the source terms are stiff uniform accuracy with respect to the stiffness parameter is a highly desirable property but it is in general very difficult to achieve using underresolved discretizations for such problems we develop different second order uniformly accurate high resolution nonoscillatory central schemes the schemes retain the simplicity of central schemes for hyperbolic conservation laws and avoid the use of riemann solvers in particular we show that these schemes possess a discrete analogue of the continuous asymptotic limit and are able to capture the correct behavior even if the initial layer and the small relaxation time are not numerically resolved

convergence of a splitting scheme applied to the ruijgrok wu model of the boltzmann equation this paper deals with upwind splitting schemes for the ruijgrok wu model physica a 113 1982 401 416 of the kinetic theory of rare7ed gases in the 8uid dynamic scaling we prove the stability and the convergence for these schemes the relaxation limit is also investigated and the limit equation is proved to be a 7rst order quasi linear conservation law the loss of quasi monotonicity of the present model makes it necessary to give a more careful analysis of its structure we also obtain global error estimates in the spaces w s p for 16s61 p 16p6 and pointwise error estimates for the approximate solution the proof naturally uses the framework introduced by nessyahu and tadmor siam j numer anal 29 1992 1505 1519 due to the convexity of the 8ux function c

knot probabilities in random diagrams we consider a natural model of random knotting choose a knot diagram at random from the finite set of diagrams with n crossings we tabulate diagrams with 10 and fewer crossings and classify the diagrams by knot type allowing us to compute exact probabilities for knots in this model as expected most diagrams with 10 and fewer crossings are unknots about 78 of the roughly 1 6 billion 10 crossing diagrams for these crossing numbers the unknot fraction is mostly explained by the prevalence of tree like diagrams which are unknots for any assignment of over under information at crossings the data shows a roughly linear relationship between the log of knot type probability and the log of the frequency rank of the knot type analogous to zipf s law for word frequency the complete tabulation and all knot frequencies are included as supplementary data

the euler lagrange equation and heat flow for the mobius energy we prove the following results 1 a unique smooth solution exists for a short time for the heat equation associated with the mobius energy of loops in a euclidean space starting with any simple smooth loop 2 a critical loop of the energy is smooth if it has cube integrable curvature combining this with an earlier result of m freedman z wang and the author we show that any local minimizer of the energy must be smooth 3 circles are the only two dimensional critical loops with cube integrable curvature the technique also applies to a family of other knot energies similar problems are open for energies of surfaces or more generally for embedded submanifolds in a fixed riemannian manifold 2000 john wiley sons inc

robustness of muscl schemes for 2d unstructured meshes we consider second order accuracy muscl schemes to approximate the solutions of hyperbolic system of conservation laws in the context of the 2d unstructured grids we propose a limitation procedure on the gradient reconstruction to enforce several stability properties we establish that the muscl scheme preserves the invariant domains and satisfy a set of entropy inequalities a conservation assumption is not useful in the present work to define the piecewise linear approximations and the proposed limitation can be understood as a systematic correction of the standard gradient reconstruction procedure the numerical method is applied to the compressible euler equations the gradient reconstruction is performed using the characteristic variables several numerical tests exhibit stability and robustness of the scheme

integral geometry of plane curves and knot invariants we study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of witten s chern simons path integral associated with a knot one of the integrals involved turns out to be a generalization of the classical crofton integral on convex plane curves and it is related with the invariants of generic plane curves recently defined by arnold with deep motivations in symplectic and contact geometry quadratic bounds on these plane curve invariants are derived using their relationship with the knot invariant

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

variants of relaxed schemes and two dimensional gas dynamics we present a relaxed scheme with more precise information about local speeds of propagation and a multidimensional construction of the cell averages the physical domain of dependence is simulated correctly and high resolution is maintained relaxation schemes have advantages that include high resolution simplicity and explicitly no approximate riemann solvers and no characteristic decomposition is necessary performance of the scheme is illustrated by tests on two dimensional euler equations of gas dynamics

balanced finite volume weno and central weno schemes for the shallow water and the open channel flow equations the goal of this work is to extend finite volume weno and central weno schemes to the hyperbolic balance laws with geometrical source term and spatially variable flux function in particular we apply proposed schemes to the shallow water and the open channel flow equations where the source term depends on the channel geometry for obtaining stable numerical schemes that are free of spurious oscillations it becomes crucial to use the decomposed source term evaluation which maintains the balancing between the flux gradient and the source term in addition the open channel flow equations contain spatially variable flux function the appropriate definitions of the terms that arise in the source term decomposition in combination with the roe approximate riemann solver that includes the spatial derivative of the flux function lead to the finite volume weno scheme that satisfies the exact conservation property the property of preserving the quiescent flow exactly when the central weno schemes are applied additional reformulations are introduced for the transition from the staggered values to the nonstaggered ones and vice versa by using the weno reconstruction procedure the proposed central weno schemes also preserve the quiescent flow but only in prismatic channels in various test problems the obtained balanced schemes show improvements in comparison with the standard versions of the proposed type schemes as well as with some other first and second order numerical schemes

an asymptotic preserving numerical scheme for kinetic equations in the low mach number limit a numerical scheme for the nonstationary boltzmann equation in the incompressible navier stokes limit is developed the scheme is induced by the asymptotic analysis of the navier stokes limit for the boltzmann equation it works uniformly for all ranges of mean free paths in the limit the scheme reduces to the chorin projection method for the incompressible navier stokes equation numerical results for different physical situations are shown and the uniform convergence of the scheme is established numerically

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

hyperbolic conservation laws with stiff relaxation terms and entropy we study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms reduced systems inviscid and viscous local conservation laws and weakly nonlinear limits are derived through asymptotic expansions an entropy condition is introduced for n n systems that ensures the hyperbolicity of the reduced inviscid system the resulting characteristic speeds are shown to be interlaced with those of the original system moreover the first correction to the reduced system is shown to be dissipative a partial converse is proved for 2 2 systems this structure is then applied to study the convergence to the reduced dynamics for the 2 2 case 1994 john wiley sons inc

on stability monotonicity and construction of difference schemes ii applications in this paper we focus on the application and illustration of the approach developed in part i this approach is found to be useful in the construction of stable and monotone central difference schemes for hyperbolic systems a new modification of the central lax friedrichs scheme is developed to be of second order accuracy the stability of several versions of the developed central scheme is proved necessary conditions for the variational monotonicity of the scheme are found a monotone piecewise cubic interpolation is used in the central schemes to give an accurate approximation for the model in question the monotonicity parameter introduced in part i is found to be important that parameter along with the courant friedrichs lewy cfl number plays a major role in the criteria for monotonicity and stability of the central schemes the modified scheme is tested on several conservation laws taking a cfl number equal or close to unity and the scheme is found to be accurate and robust

the fluid dynamic limit of the broadwell model of the nonlinear boltzmann equation in the presence of shocks this paper studies the asymptotic equivalence of the broadwell model of the nonlinear boltzmann equation to its corresponding euler equation of compressible gas dynamics in the limit of small mean free path e it is shown that the fluid dynamical approximation is valid even if there are shocks in the fluid flow although there are thin shock layers in which the convergence does not hold more precisely by assuming that the fluid solution is piecewise smooth with a finite number of noninteracting shocks and suitably small oscillations we can show that there exist solutions to the broadwell equations such that the broadwell solutions converge to the fluid dynamical solutions away from the shocks at a rate of order e as the mean free path e goes to zero for the proof we first construct a formal solution for the broadwell equation by matching the truncated hilbert expansion and shock layer expansion then the existence of broadwell solutions and its convergence to the fluid dynamic solution is reduced to the stability analysis for the approximate solution we use an energy method which makes full use of the inner structure of time dependent shock profiles for the broadwell equations

on a class of uniformly accurate imex runge kutta schemes and applications to hyperbolic systems with relaxation in this paper we consider hyperbolic systems with relaxation in which the relaxation time varepsilon may vary from values of order one to very small values when varepsilon is very small the relaxation term becomes very strong and highly stiff and underresolved numerical schemes may produce spurious results in such cases it is important to have schemes that work uniformly with respect to varepsilon implicit explicit imex runge kutta r k schemes have been widely used for the time evolution of hyperbolic partial differential equations but the schemes existing in literature do not exhibit uniform accuracy with respect to the relaxation time we develop new imex r k schemes for hyperbolic systems with relaxation that present better uniform accuracy than the ones existing in the literature and in particular produce good behavior with high order accuracy in the asymptotic limit i e when varepsilon is very small these schemes are obtained by imposing new additional order conditions to guarantee better accuracy over a wide range of the relaxation time we propose the construction of new third order imex r k schemes of type ck s boscarino siam j numer anal 45 2008 pp 1600 1621 in several test problems these schemes with a fixed spatial discretization exhibit for all range of the relaxation time an almost uniform third order accuracy

a hybrid method for hydrodynamic kinetic flow part i a particle grid method for reducing stochastic noise in kinetic regimes in this work we present a hybrid particle grid monte carlo method for the boltzmann equation which is characterized by a significant reduction of the stochastic noise in the kinetic regime the hybrid method is based on a first order splitting in time to separate the transport from the relaxation step the transport step is solved by a deterministic scheme while a hybrid dsmc based method is used to solve the collision step such a hybrid scheme is based on splitting the solution in a collisional and a non collisional part at the beginning of the collision step and the dsmc method is used to solve the relaxation step for the collisional part of the solution only this is accomplished by sampling only the fraction of particles candidate for collisions from the collisional part of the solution performing collisions as in a standard dsmc method and then projecting the particles back onto a velocity grid to compute a piecewise constant reconstruction for the collisional part of the solution the latter is added to a piecewise constant reconstruction of the non collisional part of the solution which in fact remains unchanged during the relaxation step numerical results show that the stochastic noise is significantly reduced at large knudsen numbers with respect to the standard dsmc method indeed in this algorithm the particle scheme is applied only on the collisional part of the solution so only this fraction of the solution is affected by stochastic fluctuations but since the collisional part of the solution reduces as the knudsen number increases stochastic noise reduces as well at large knudsen numbers

large time step and asymptotic preserving numerical schemes for the gas dynamics equations with source terms we propose large time step and asymptotic preserving schemes for the gas dynamics equations with external forces and friction terms by asymptotic preserving we mean that the numerical scheme is able to reproduce at the discrete level the parabolic type asymptotic behavior satisfied by the continuous equations by large time step we mean that the scheme is stable under a cfl stability condition driven by the slow material waves and not by the fast acoustic waves customary in godunov type schemes numerical evidence is proposed and shows a gain of several orders of magnitude in both accuracy and efficiency

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

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

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

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

design and implementation of predictors for additive semi implicit runge kutta methods space discretization of some time dependent partial differential equations gives rise to stiff systems of ordinary differential equations in this case implicit methods should be used and therefore in general nonlinear systems must be solved the solutions to these systems are approximated by iterative schemes and in order to obtain an efficient code good initializers should be used recently a parallel code based on some runge kutta and additive runge kutta methods has been constructed focusing especially on additive semi implicit runge kutta asirk methods the aim of the present paper is to develop efficient initializers for these methods

uniform stability of a finite difference scheme for transport equations in diffusive regimes an asymptotic preserving numerical scheme with respect to diffusion scalings for a linear transport equation is investigated the scheme is adopted from a class of schemes developed in s jin l pareschi and g toscani siam j numer anal 38 2000 pp 913 936 and a klar siam j numer anal 35 1998 pp 1073 1094 stability is proven uniformly in the mean free path under a cfl type condition turning into a parabolic cfl condition in the diffusion limit

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

staggered mixed finite volume finite element method for the navier stokes equations to solve flow problems associated with the navier stokes equations we construct a mixed finite volume finite element method for the spatial approximation of the convective and diffusive parts of the flux respectively the finite volume component of the method is adapted from the authors construction for hyperbolic conservation laws and rectangular or unstructured triangular grids of two dimensional finite volume extensions of the lax friedrichs and nessyahu tadmor central difference schemes in which the resolution of riemann problems at cell interfaces is bypassed thanks to the use of the lax friedrichs scheme on two specific staggered grids muscl type piecewise linear cell interpolants slope limiters and a two step predictor corrector time discretization lead to an oscillation free quasi second order resolution for the viscous terms we use a centered finite element approximation inspired by rostand stoufflet to improve the quality of the resolution we use a grid adaptation algorithm proposed by castro diaz and hecht

central schemes for balance laws of relaxation type several models in mathematical physics are described by quasi linear hyperbolic systems with source term and in several cases the production term can become stiff here suitable central numerical schemes for such problems are developed and applications to the broadwell model and extended thermodynamics are presented the numerical methods are a generalization of the nessyahu tadmor scheme to the nonhomogeneous case by including the cell averages of the production terms in the discrete balance equations a second order scheme uniformly accurate in the relaxation parameter is derived and its properties analyzed numerical tests confirm the accuracy and robustness of the scheme

numerical methods for hyperbolic conservation laws with stiff relaxation i spurious solutions the author considers the numerical solution of hyperbolic systems of conservation laws with relaxation using a shock capturing finite difference scheme on a fixed uniform spatial grid it is conjectured that certain a priori criteria ensure that the numerical method does not produce spurious solutions as the relaxation time vanishes one criterion is that the limits of vanishing relaxation time and vanishing viscosity commute for the viscous regularization of the hyperbolic system a second criterion is that a certain subcharacteristic condition be satisfied by the hyperbolic system this conjecture is supported with analytical and numerical results for a specific example the solution of generalized riemann problems of a model system of equations with a fractional step scheme in which godunov s method is coupled with the backward euler method similar ideas are applied to the numerical solution of stiff detonation problems

asymptotic preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings in this paper we develop a set of stochastic numerical schemes for hyperbolic and transport equations with diffusive scalings and subject to random inputs the schemes are asymptotic preserving ap in the sense that they preserve the diffusive limits of the equations in discrete setting without requiring excessive refinement of the discretization our stochastic ap schemes are extensions of the well developed deterministic ap schemes to handle the random inputs we employ generalized polynomial chaos gpc expansion and combine it with stochastic galerkin procedure we apply the gpc galerkin scheme to a set of representative hyperbolic and transport equations and establish the ap property in the stochastic setting we then provide several numerical examples to illustrate the accuracy and effectiveness of the stochastic ap schemes

asymptotic preserving well balanced schemes for radiative transfer and the rosseland approximation we are concerned with efficient numerical simulation of the radiative transfer equations to this end we follow the well balanced approach s canvas and reformulate the relaxation term as a nonconservative product regularized by steady state curves while keeping the velocity variable continuous these steady state equations are of fredholm type the resulting upwind schemes are proved to be stable under a reasonable parabolic cfl condition of the type t o x2 among other desirable properties some numerical results demonstrate the realizability and the efficiency of this process

an analysis of operator splitting techniques in the stiff case operator splitting methods are commonly used in many applications we focus here on the case where the evolution equations to be simulated are stiff we will more particularly consider the case of two operators a stiff one and a nonstiff one this occurs in numerous application fields e g combustion air pollution and reactive flows the classical analysis of the splitting error may then fail since the chosen splitting timestep t is in practice much larger than the fastest time scales the asymptotic expansion t 0 is therefore no longer valid we show here that singular perturbation theory provides an interesting framework for the study of splitting error some new results concerning the order of local errors are derived the main result deals with the choice of the sequential order for the operators the stiff operator must always be last in the splitting scheme

a micro macro decomposition based asymptotic preserving scheme for the multispecies boltzmann equation in this paper we extend the micro macro decomposition based asymptotic preserving scheme developed by bennoune lemou and mieussens j comput phys 227 2008 pp 3781 3803 for the single species boltzmann equation to the multispecies problems an asymptotic preserving scheme for the kinetic equation is very efficient in the fluid regime where the knudsen number is small and the collision term becomes stiff it allows a coarse independent of the knudsen number mesh size and a large time step in the fluid regime the difficulty associated with multispecies problems is that there are no local conservation laws for each species resulting in extra stiff nonlinear source terms that need to be discretized properly in order to 1 avoid newton type solvers for nonlinear algebraic systems and 2 to be asymptotic preserving we show that these extra nonlinear source terms can be solved using only linear system solvers and the scheme preserves the correct euler and navier stokes limits numerical examples are used to demonstrate the efficiency and applicability of the schemes for both euler and navier stokes regimes

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

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

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

the degenerate analogue of ariki s categorification theorem we explain how to deduce the degenerate analogue of ariki s categorification theorem over the ground field mathbb c as an application of schur weyl duality for higher levels and the kazhdan lusztig conjecture in finite type a we also discuss some supplementary topics including young modules tensoring with sign tilting modules and ringel duality

an asymptotic preserving scheme for hydrodynamics radiative transfer models numerics for radiative transfer in this paper we present a numerical scheme for a hydrodynamics radiative transfer model consisting of two steps the first one is based on a relaxation method and the second one on the well balanced scheme the derivation of the scheme relies on the resolution of a stationary riemann problem with source terms the obtained scheme preserves the limited flux property and it is compatible with the diffusive regime of hydrodynamics radiative transfer models these properties are illustrated by numerical tests one of them involves a radiative transfer model coupled with an equation for the temperature of the material

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

a numerical study of the axisymmetric couette taylor problem using a fast high resolution second order central scheme we present a numerical study of the axisymmetric couette taylor problem using a finite difference scheme the scheme is based on a staggered version of a second order central differencing method combined with a discrete hodge projection the use of central differencing operators obviates the need to trace the characteristic flow associated with the hyperbolic terms the result is a simple and efficient scheme which is readily adaptable to other geometries and to more complicated flows the scheme exhibits competitive performance in terms of accuracy resolution and robustness the numerical results agree accurately with linear stability theory and with previous numerical studies

error analysis of imex runge kutta methods derived from differential algebraic systems in this paper we present an error analysis of the imex runge kutta methods when applied to stiff problems containing a nonstiff term and a stiff term characterized by a small stiffness parameter varepsilon in this analysis we expand the global error in powers of varepsilon and show that the coefficients of the error are the global errors of the imex runge kutta method applied to a differential algebraic system interesting convergence results of these errors and of the remainder of the expansion allow us to determine sharp error bounds for stiff problems as a representative example of stiff problems we have chosen the van der pol equation we illustrate that the theoretical prediction is confirmed by the numerical test specifically an order reduction phenomenon is observed when the problem becomes increasingly stiff in particular making several assumptions we try to improve global error estimates of several imex runge kutta methods existing in the literature

extrapolation methods for hyperbolic systems with relaxation abstract a splitting scheme is used for numerical solution of hyperbolic systems with a stiff relaxation high order flux in the convection step guarantees a good spatial accuracy the relaxation term is treated implicity extrapolation is used to combine the steps obtaining high accuracy in time a third order scheme both in space and in time is presented and applied to the broadwell model of gas dynamics

degenerate cyclotomic hecke algebras and higher level heisenberg categorification abstract we associate a monoidal category h to each dominant integral weight of sl p or sl these categories defined in terms of planar diagrams act naturally on categories of modules for the degenerate cyclotomic hecke algebras associated to we show that in the sl case the level d heisenberg algebra embeds into the grothendieck ring of h where d is the level of the categories h can be viewed as a graphical calculus describing induction and restriction functors between categories of modules for degenerate cyclotomic hecke algebras together with their natural transformations as an application of this tool we prove a new result concerning centralizers for degenerate cyclotomic hecke algebras

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

the effects of slope limiting on asymptotic preserving numerical methods for hyperbolic conservation laws many hyperbolic systems of equations with stiff relaxation terms reduce to a parabolic description when relaxation dominates an asymptotic preserving numerical method is a discretization of the hyperbolic system that becomes a valid discretization of the parabolic system in the asymptotic limit we explore the consequences of applying a slope limiter to the discontinuous galerkin dg method with linear elements for hyperbolic systems with stiff relaxation terms without a limiter the dg method gives an accurate discretization of the chapman enskog approximation of the system when the relaxation length scale is not resolved it is well known that the first order upwind or step method fails to obtain the proper asymptotic limit we show that using the minmod slope limiter also fails but that using double minmod gives the proper asymptotic limit despite its effectiveness in the asymptotic limit the double minmod limiter allows artificial extrema at cell interfaces referred to as sawteeth we present a limiter that eliminates the sawteeth but maintains the proper asymptotic limit the systems that we analyze are the hyperbolic heat equation and the p n thermal radiation equations numerical examples are used to verify our analysis

computation of nonclassical solutions to hamilton jacobi problems this paper is devoted to the construction of numerical methods for the approximation of nonclassical solutions to multidimensional hamilton jacobi equations for both scalar and vectorial problems recent theoretical results have yielded existence of solutions in many cases for which the usual viscosity approach was ill suited or not applicable the selection criterion used here is based on a viscoelasticity capillarity approach common in solid mechanics numerical methods adapted to this framework are built consistency of the model equation with the given selection criterion is essential it is achieved here through the use of high order finite difference schemes by considering applications to potential well problems the convergence of the methods are investigated

on the self linking of knots this note describes a subcomplex f of the de rham complex of parametrized knot space which is combinatorial over a number of universal anomaly integrals the self linking integrals of guadaguini martellini and mintchev perturbative aspects of chern simons field theory phys lett b 227 111 1989 and bar natan perturbative aspects of the chern simons topological quantum field theory ph d thesis princeton university 1991 also on the vassiliev knot invariants to appear in topology are seen to represent the first nontrivial element in h0 f occurring at level 4 and are anomaly free however already at the next level an anomalous term is possible