localization effects and measure source terms in numerical schemes for balance laws this paper investigates the behavior of numerical schemes for non linear conservation laws with source terms we concentrate on two significant examples relaxation approximations and genuinely nonhomogeneous scalar laws the main tool in our analysis is the extensive use of weak limits and nonconservative products which allow us to describe accurately the operations achieved in practice when using riemann based numerical schemes some illustrative and relevant computational results are provided

global bv entropy solutions and uniqueness for hyperbolic systems of balance laws we consider the cauchy problem for n n strictly hyperbolic systems of nonresonant balance laws

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

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

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

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

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

identification of asymptotic decay to self similarity for one dimensional filtration equations the objective of this paper is the derivation and the analysis of a simple explicit numerical scheme for general one dimensional filtration equations it is based on an alternative formulation of the problem using the pseudoinverse of the density s repartition function in particular the numerical approximations can be proven to satisfy a contraction property for a wasserstein metric various numerical results illustrate the ability of this numerical process to capture the time asymptotic decay towards self similar solutions even for fast diffusion equations

time asymptotic high order schemes for dissipative bgk hyperbolic systems we introduce a new class of finite differences schemes to approximate one dimensional dissipative semilinear hyperbolic systems with a bgk structure using precise analytical time decay estimates of the local truncation error it is possible to design schemes based on the standard upwind approximation which are increasingly accurate for large times when approximating small perturbations of constant asymptotic states numerical tests show their better performances with respect to those of other schemes

transient l1 error estimates for well balanced schemes on non resonant scalar balance laws abstract the ability of well balanced wb schemes to capture very accurately steady state regimes of non resonant hyperbolic systems of balance laws has been thoroughly illustrated since its introduction by greenberg and leroux 1996 15 see also the anterior wb glimm scheme in e 1992 8 this paper aims at showing by means of rigorous c t 0 l x 1 estimates that these schemes deliver an increased accuracy in transient regimes too namely after explaining that for the vast majority of non resonant scalar balance laws the c t 0 l x 1 error of conventional fractional step tang and teng 1995 45 numerical approximations grows exponentially in time like exp max g t x as a consequence of the use of gronwall s lemma it is shown that wb schemes involving an exact riemann solver suffer from a much smaller error amplification thanks to strict hyperbolicity their error grows at most only linearly in time see also layton 1984 30 numerical results on several test cases of increasing difficulty including the classical leveque yee s benchmark problem leveque and yee 1990 34 in the non stiff case confirm the analysis

maxwellian decay for well balanced approximations of a super characteristic chemotaxis model we focus on the numerical simulation of a one dimensional model of chemotaxis dynamics proposed by greenberg and alt trans amer math soc 300 1987 pp 235 258 in a bounded domain by means of a previously introduced well balanced wb and asymptotic preserving ap scheme l gosse j math anal appl 2011 we are especially interested in studying the decay onto numerical steady states for two reasons 1 conventional upwind schemes have been shown to stabilize onto spurious non maxwellian states with a very big mass flux see e g f guarguaglini et al discrete contin dyn syst ser b 12 2009 pp 39 76 and 2 the initial data lead to a dynamic which is mostly super characteristic in the sense of s jin and m katsoulakis siam j appl math 61 2000 pp 273 292 thus the stability results of gosse do not apply a reflecting boundary condition which is compatible with the well balanced character is presented a mass preservation property is proved and some results on super characteristic relaxation are recalled numerical experiments with coarse computational grids are presented in detail they illustrate the bifurcation diagrams of guarguaglini et al which relate the total initial mass of cells with the time asymptotic values of the chemoattractant substance on each side of the domain it is shown that the wb scheme stabilizes correctly onto zero mass flow rate hence maxwellian steady states which agree with the aforementioned bifurcation diagrams the evolution in time of residues is commented for every considered test case

asymptotic preserving scheme for the shallow water equations with source terms on unstructured meshes the following work is devoted to the construction and validation of a numerical scheme for the 2d shallow water system on unstructured meshes supplemented by topography and friction source terms approximate solutions of frictionless flows are obtained considering a suitable formulation of the conservation laws involving the water free surface and some fractions of water accounting for the topography variations the discretization of the friction source terms relies on the use of a modified riemann solver for the flux computation the resulting scheme is then corrected in order to achieve an asymptotic regime preservation a muscl reconstruction is also performed to increase the space order of accuracy the overall numerical approach is shown to be consistent well balanced and to satisfy a domain invariant principle these results are assessed through several benchmark tests involving complex geometry and varying bathymetry in the presence of dry areas special interest is given to the wave front speed computation highlighting the stability of the method even when implementing the asymptotic preserving correction

discrete velocity models of the boltzmann equation a survey on the mathematical aspects of the theory discrete velocity models of the boltzmann equation are of considerable conceptual interest in the kinetic theory of gases and at the same time a fascinating mathematical subject the last decade has seen a lot of work on this subject which is reviewed here we put an emphasis on mathematical aspects in particular the global existence question qualitative properties of solutions explicit solutions and the fluid dynamical limit the mechanical background is discussed more concisely in part i numerous contributions concerning other questions are outlined we try to give a complete reference list up to and including 1985

stability of godunov s method for a class of 2 2 systems of conservation laws on demontre la stabilite et la convergence du schema de godunov pour une classe particuliere de systemes 2 2 non lineaires de lois de conservation

heegner divisors l functions and harmonic weak maass forms recent works mostly related to ramanujan s mock theta functions make use of the fact that harmonic weak maass forms can be combinatorial generating functions generalizing works of waldspurger kohnen and zagier we prove that such forms also serve as generating functions for central values and derivatives of quadratic twists of weight 2 modular l functions to obtain these results we construct differentials of the third kind with twisted heegner divisor by suitably generalizing the borcherds lift to harmonic weak maass forms the connection with periods fourier coefficients derivatives of l functions and points in the jacobian of modular curves is obtained by analyzing the properties of these differentials using works of scholl waldschmidt and gross and zagier

a well balanced scheme for the numerical processing of source terms in hyperbolic equations in a variety of physical problems one encounters source terms that are balanced by internal forces and this balance supports multiple steady state solutions that are stable typical of these are gravity driven flows such as those described by the shallow water equations over a nonuniform ocean bottom 1 10 qquad h t hu x 0 text and hu t left hu 2 frac gh 2 2 right x ga x x h 0 many classic numerical schemes cannot maintain these steady solutions or achieve them in the long time limit with an acceptable level of accuracy because they do not preserve the proper balance between the source terms and internal forces we propose here a numerical scheme adapted to a scalar conservation law that preserves this balance and that can it is hoped be extended to more general hyperbolic systems the proof of convergence of this scheme toward the entropy solution is given and some numerical tests are reported

on the asymptotic properties of imex runge kutta schemes for hyperbolic balance laws implicit explicit imex schemes are a powerful tool in the development of numerical methods for hyperbolic systems with stiff sources here we focus our attention on the asymptotic properties of such schemes like the preservation of steady states well balanced property and the behavior in presence of small space time scales asymptotic preservation property we analyze conditions under which the standard additive approach based on taking the fluxes explicitly and the sources implicitly yields a well balanced behavior in addition we consider a partitioned strategy which possesses better well balanced properties the behavior of the additive and partitioned approaches under classical scaling limits is then studied in the context of asymptotic preserving schemes additional order conditions that guarantee the correct behavior of the schemes in the navier stokes regime are derived several examples illustrate these asymptotic behaviors and the performance of the new methods

a completely solvable model of the nonlinear boltzmann equation abstract in one space and one time dimension a model of the nonlinear boltzmann equation is presented that is exactly solvable for all initial conditions furthermore this model has the following desirable properties i conservation of the number of particles ii energy conservation iii nonlinearity iv positivity of distribution functions and v unique equilibrium state for any given density which is approached as t for most physically interesting initial conditions some of the simple properties of this model are studied

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

a high order asymptotic preserving scheme for kinetic equations using projective integration we investigate a high order fully explicit asymptotic preserving scheme for a kinetic equation with linear relaxation both in the hydrodynamic and diffusive scalings in which a hyperbolic resp parabolic limiting equation exists the scheme first takes a few small inner steps with a simple explicit method such as direct forward euler to damp out the stiff components of the solution and estimate the time derivative of the slow components these estimated time derivatives are then used in an outer runge kutta method of arbitrary order we show that with an appropriate choice of inner step size the time step restriction on the outer time step is similar to the stability condition for the limiting macroscopic equation moreover the number of inner time steps is also independent of the scaling parameter we analyze stability and consistency and illustrate with numerical results

a semiclassical coupled model for the transient simulation of semiconductor devices we consider the approximation of a microelectronic device corresponding to an n n n diode consisting of a channel flanked on both sides by two highly doped regions this is modeled through a system of equations ballistic for the channel and drift diffusion elsewhere the overall coupling stems from the poisson equation for the self consistent potential we propose an original numerical method for its processing being realizable explicit in time and nonnegativity preserving on the density in particular the boundary conditions at the junctions express the continuity of the current and do not destabilize the general scheme efficiency is shown by presenting results on test cases of some practical interest

asymptotic preserving and well balanced schemes for the 1d cattaneo model of chemotaxis movement in both hyperbolic and diffusive regimes abstract the original well balanced wb framework of greenberg and leroux 1996 24 and gosse 2002 18 relying on non conservative nc products see lefoch and tzavaras 1999 41 is set up in order to efficiently treat the so called cattaneo model of chemotaxis in 1d see hillen and stevens 2000 29 it proceeds by concentrating the source terms onto dirac masses this allows to handle them by nc jump relations based on steady state equations which can be integrated explicitly a riemann solver is deduced and the corresponding wb godunov scheme completed with the standard hoff smoller theory see hillen and stevens 2000 29 for the diffusion reaction equation ruling the evolution of the chemoattractant concentration is studied in detail later following former results of gosse and toscani 2002 2003 21 22 a simple rewriting of the nc jump relations allows to generate another version of the same godunov scheme which is well adapted to the parabolic scaling involving a small parameter e the standard bv framework is used to study the uniform stability of this asymptotic preserving ap scheme with respect to e allows to pass to the limit and derive a simple centered discretization of the keller segel model finally results by filbet 2006 14 permit to pass to the complementary limit when the space step h is sent to zero numerical results are included to illustrate the feasibility and the efficiency of the method

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

asymptotic high order mass preserving schemes for a hyperbolic model of chemotaxis we introduce a new class of finite difference schemes for approximating the solutions to an initial boundary value problem on a bounded interval for a one dimensional dissipative hyperbolic system with an external source term which arises as a simple model of chemotaxis since the solutions to this problem may converge to nonconstant asymptotic states for large times standard schemes usually fail to yield a good approximation therefore we propose a new class of schemes which use an asymptotic higher order correction second and third order in our examples to balance the effects of the source term and the influence of the asymptotic solutions special care is needed to deal with boundary conditions to avoid harmful loss of mass convergence results are proved for these new schemes and several numerical tests are presented and discussed to verify the effectiveness of their behavior

hydrodynamic limits for kinetic equations and the diffusive approximation of radiative transport for acoustic waves we consider a class of kinetic equations equipped with a single conservation law which generate l 1 contractions we discuss the hydrodynamic limit to a scalar conservation law and the diffusive limit to a possibly degenerate parabolic equation the limits are obtained in the dissipative sense equivalent to the notion of entropy solutions for conservation laws which permits the use of the perturbed test function method and allows for simple proofs a general compactness framework is obtained for the diffusive scaling in l 1 the radiative transport equations satisfied by the wigner function for random acoustic waves present such a kinetic model that is endowed with conservation of energy the general theory is used to validate the diffusive approximation of the radiative transport equation

convergence of relaxation schemes to the equations of elastodynamics we study the effect of approximation matrices to semi discrete relaxation schemes for the equations of one dimensional elastodynamics we consider a semi discrete relaxation scheme and establish convergence using the l p theory of compensated compactness then we study the convergence of an associated relaxation diffusion system inspired by the scheme numerical comparisons of fully discrete schemes are carried out

space localization and well balanced schemes for discrete kinetic models in diffusive regimes we derive and study well balanced schemes for quasi monotone discrete kinetic models by means of a rigorous localization procedure we reformulate the collision terms as nonconservative products and solve the resulting riemann problem whose solution is self similar the construction of an asymptotic preserving ap godunov scheme is straightforward and various compactness properties are established within different scalings finally some computational results are supplied to show that this approach is realizable and efficient on concrete 2 2 models

the convergence of numerical transfer schemes in diffusive regimes i discrete ordinate method in highly diffusive regimes the transfer equation with anisotropic boundary conditions has an asymptotic behavior as the mean free path epsilon tends to zero that is governed by a diffusion equation and boundary conditions obtained through a matched asymptotic boundary layer analysis a numerical scheme for solving this problem has an epsilon 1 contribution to the truncation error that generally gives rise to a nonuniform consistency with the transfer equation for small epsilon thus degrading its performance in diffusive regimes in this paper we show that whenever the discrete ordinate method has the correct diffusion limit both in the interior and at the boundaries its solutions converge to the solution of the transport equation uniformly in epsilon our proof of the convergence is based on an asymptotic diffusion expansion and requires error estimates on a matched boundary layer approximation to the solution of the discrete ordinate method

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

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

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

asymptotic preserving and positive schemes for radiation hydrodynamics in view of radiation hydrodynamics computations we propose an implicit and positive numerical scheme that captures the diffusion limit of the two moments approximate model for the radiative transfer even on coarses grids the positivity of the scheme is equivalent to say that the scheme preserves the limited flux property various test cases show the accuracy and robustness of the scheme

the utility of mixed form likelihoods summary we highlight a feature of likelihood based methods that provides flexibility in model formulation and inference in particular overall likelihoods that consist of likelihood contributions with different forms are considered the particular forms may be predetermined by design criteria or may be selected based on features of the data inferences based on such mixed form likelihoods are valid provided standard regularity conditions hold and the parameters of interest have the same interpretation in the various forms the advantages of constructing overall likelihoods in this way are illustrated by applications involving the analysis of 2 2 k tables and left censored water quality data

a kinetic equation with kinetic entropy functions for scalar conservation laws we construct a nonlinear kinetic equation and prove that it is welladapted to describe general multidimensional scalar conservation laws in particular we prove that it is well posed uniformly in e the microscopic scale we also show that the proposed kinetic equation is equipped with a family of kinetic entropy functions analogous to boltzmann s microscopich function such that they recover krushkov type entropy inequality on the macroscopic scale finally we prove by both bv compactness arguments in the multidimensional case and by compensated compactness arguments in the one dimensional case that the local density of kinetic particles admits a continuum limit as it converges strongly with e 0 to the unique entropy solution of the corresponding conservation law