lyapunov function for the kuramoto model of nonlinearly coupled oscillators a lyapunov function for the phase locked state of the kuramoto model of non linearly coupled oscillators is presented it is also valid for finite range interactions and allows the introduction of thermodynamic formalism such as ground states and universality classes for the kuramoto model a minimum of the lyapunov function corresponds to a ground state of a system with frustration the interaction between the oscillators xy spins is ferromagnetic whereas the random frequencies induce random fields which try to break the ferromagnetic order i e global phase locking the ensuing arguments imply asymptotic stability of the phase locked state up to degeneracy and hold for any probability distribution of the frequencies special attention is given to discrete distribution functions we argue that in this case a perfect locking on each of the sublattices which correspond to the frequencies results but that a partial locking of some but not all sublattices is not to be expected the order parameter of the phase locked state is shown to have a strictly positive lower bound r 1 2 so that a continuous transition to a nonlocked state with vanishing order parameter is to be excluded

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

the unique limit of the glimm scheme we introduce the definitions of a standard riemann semigroup and of a viscosity solution for a nonlinear hyperbolic system of conservation laws for a class including general 2 2 systems it is proved that the solutions obtained by a wavefront tracking algorithm or by the glimm scheme are precisely the semigroup trajectories in particular these solutions are unique and depend lipschitz continuously on the initial data in the l1 norm

on relaxation hyperbolic systems violating the shizuta kawashima condition in this paper we start a general study on relaxation hyperbolic systems which violate the shizuta kawashima sk coupling condition this investigation is motivated by the fact that this condition is not satisfied by various physical systems and almost all the time in several space dimensions first we explore the role of entropy functionals around equilibrium solutions which may not be constant proposing a stability condition for such solutions then we find strictly dissipative entropy functions for one dimensional 2 2 systems which violate the sk condition finally we prove the existence of global smooth solutions for a class of systems such that condition sk does not hold but which are linearly degenerated in the non dissipative directions

convergence of a relaxation scheme for hyperbolic systems of conservation laws this paper concerns the study of a relaxation scheme for n times n hyperbolic systems of conservation laws in particular with the compensated compactness techniques we prove a rigorous result of convergence of the approximate solutions toward an entropy solution of the equilibrium system as the relaxation time and the mesh size tend to zero

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

a high order relaxation method with projective integration for solving nonlinear systems of hyperbolic conservation laws we present a general high order fully explicit relaxation scheme which can be applied to any system of nonlinear hyperbolic conservation laws in multiple dimensions the scheme consists of two steps in a first relaxation step the nonlinear hyperbolic conservation law is approximated by a kinetic equation with stiff bgk source term then this kinetic equation is integrated in time using a projective integration method after taking a few small inner steps with a simple explicit method such as direct forward euler to damp out the stiff components of the solution the time derivative is estimated and used in an outer rungekutta 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 cfl condition for the hyperbolic conservation law moreover the number of inner time steps is also independent of the stiffness of the bgk source term we discuss stability and consistency and illustrate with numerical results linear advection burgers equation and the shallow water and euler equations in one and two spatial dimensions

global bv solutions to a p system with relaxation in this paper we study the global existence of bv solutions to a p system with a relaxation source term using the glimm s scheme the pressure p is given by a law with 1 by a suitable choice of the measure for the strength of the shock waves we show that the total strength of the waves and the total variation of the solutions are uniformly bounded with respect to the relaxation parameter furthermore when the relaxation parameter tends to zero we show that the sequence of bv solutions eventually converges to a weak solution to the equilibrium equation 2000 academic press

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 convergent scheme for hamilton jacobi equations on a junction application to traffic in this paper we consider first order hamilton jacobi hj equations posed on a junction that is to say the union of a finite number of half lines with a unique common point for this continuous hj problem we propose a finite difference scheme and prove two main results as a first result we show bounds on the discrete gradient and time derivative of the numerical solution our second result is the convergence for a subsequence of the numerical solution towards a viscosity solution of the continuous hj problem as the mesh size goes to zero when the solution of the continuous hj problem is unique we recover the full convergence of the numerical solution we apply this scheme to compute the densities of cars for a traffic model we recover the well known godunov scheme outside the junction point and we give a numerical illustration

nonlinear stability of shock fronts for a relaxation system in several space dimensions it is proposed by jin and xin in 3 to use the following relaxation system u t sum limits i 1 m left upsilon i right x i 0 text u in r n text upsilon i in r n 1 upsilon it a i u x i frac left upsilon i f i left u right right varepsilon text i 1 2 ldots m 2 to approximate multi dimensional hyperbolic conservation laws of the form u t sum limits i 1 m f i left u right x i 0 left x i t right in r 1 times r u in r n 3 here e is a positive constant representing the rate of relaxation and a i a i i a i 0 i 1 2 m with i being the n n identity matrix and a i being positive constants satisfying the subcharacteristic condition see 3 and sc one of the main advantages of the system 1 2 is its form of local relaxation structure and linearity in convection which makes it possible to solve this system quite easily by underresolved stable numerical discretizations using neither riemann solvers spatially nor nonlinear system of algebraic equations solvers temporally a class of numerical schemes solving 1 2 has been proposed in 3 which yields satisfactory numerical solutions to hyperbolic conservation laws 3 3 yet the asymptotic equivalence of the relaxation system 1 2 and the hyperbolic conservation laws 3 as either the rate of relaxation goes to zero or time approaches infinity has been rigorously analysed only in one space dimension see 5 9 and 11

a matrix valued kuramoto model the kuramoto model is an important model for studying the onset of phase locking in an ensemble of nonlinearly coupled phase oscillators each oscillator has a natural frequency often taken to be random and interacts with the other oscillators through the phase difference it is known that as the coupling strength is increased there is a bifurcation in which the incoherent state becomes unstable and a stable phase locked solution is born beginning with the work of lohe there have been a number of paper that have generalized the kuramoto model for phase locking to a non commuting setting here we propose and analyze another such model we consider a collection of matrix valued variables that evolve in such a way as to try to align their eigenvector frames the phase locked state is one where the eigenframes all align and thus the matrices all commute we analyze the stability of the equal frequency phase locked state and show that it is stable we also analyze for the case of symmetric matrices a dynamic analog of the twist states arising in the standard kuramoto model and show that these twist states are dynamically unstable

time discretizations for numerical optimisation of hyperbolic problems abstract we consider a class of numerical schemes for optimal control problems of hyperbolic conservation laws we focus on finite volume schemes using relaxation as a numerical approach to the optimality system in particular we study the arising numerical schemes for the adjoint equation and derive necessary conditions on the time integrator we show that the resulting schemes are in particular asymptotic preserving for both the adjoint and forward equation we furthermore prove that higher order time integrator yields suitable runge kutta schemes the discussion includes the numerically interesting zero relaxation case

zero relaxation limit to centered rarefaction waves for a rate type viscoelastic system abstract we study a rate type viscoelastic system proposed by i suliciu 1990 internat j engrg sci 28 827 841 which is a 3 3 hyperbolic system with relaxation as the relaxation time tends to zero this system convergences to the well known p system formally in the case where the initial data are the riemann data such that the corresponding solutions of the p system are centered rarefaction waves we show that if the wave strength is suitably small then the solution for the relaxation system exists globally in time and converges to the solution of the corresponding rarefaction waves uniformly as the relaxation time goes to zero except for an initial layer the jump discontinuities in the solutions are decaying exponentially fast as time tends to infinity

asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy we study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in m space dimensions under the shizuta kawashima condition we show that these solutions approach constant equilibrium state in the lp norm at a rate o t m 2 1 1 p as t tends to infty for p in min m 2 infty moreover we can show that we can approximate with a faster order of convergence theconservative part of the solution in terms of the linearized hyperbolic operator for m 2 and by a parabolic equation in the spirit of chapman enskog expansion the main tool is given by a detailed analysis of the green function for the linearized problem

approximation of the multi dimensional incompressible navier stokes equations by discrete velocity vector bgk models abstract in this paper we prove that a class of discrete velocity vector bgk models can be used as approximations to the multi dimensional incompressible navier stokes equations on the torus the proof relies crucially on two constructions a new symmetrizer and approximate solutions the former renders us a natural setting for energy estimates and the latter is motivated by the maxwell iteration simple and concise stability conditions are also obtained for a number of concrete 2 or 3 dimensional bgk models

asymptotic stability of planar rarefaction waves for the relaxation approximation of conservation laws in several dimensions abstract this paper concerns the large time behavior toward planar rarefaction waves of solutions for the relaxation approximation of conservation laws in several dimensions it is shown that a planar rarefaction wave is nonlinear stable in the sense that it is an asymptotic attractor for the relaxation approximation of conservation laws

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

stability and convergence of a class of finite element schemes for hyperbolic systems of conservation laws we propose a class of finite element schemes for systems of hyperbolic conservation laws that are based on finite element discretizations of appropriate relaxation models we consider both semidiscrete and fully discrete finite element schemes and show that the schemes are stable and when the compensated compactness theory is applicable do converge to a weak solution of the hyperbolic system the schemes use piecewise polynomials of arbitrary degree and their consistency error is of high order we also prove that the rate of convergence of the relaxation system to a smooth solution of the conservation laws is of order o eps

existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in eulerian and lagrangian coordinates we prove the existence and compactness stability of entropy solutions for the hyperbolic systems of conservation laws corresponding to the isentropic gas dynamics where the pressure and density are related by a law for any 1 our results considerably extend and simplify the program initiated by diperna and provide a complete existence proof our methods are based on the compensated compactness and the kinetic formulation of systems of conservation laws 1996 john wiley sons inc

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

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

the lip stability and error estimates for a relaxation scheme we show the discrete lip stability for a relaxation scheme proposed by jin and xin comm pure appl math 48 1995 pp 235 277 to approximate convex conservation laws equipped with the lip stability we obtain global error estimates in the spaces w s p for 1 leq s leq 1 p 1 leq p leq infty and pointwise error estimates for the approximate solution obtained by the relaxation scheme the proof uses the framework introduced by nessyahu and tadmor siam j numer anal 29 1992 pp 1505 1519 we also show a maximum principle for the relaxation scheme when the initial data are in an equilibrium state

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

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

asymptotic numerical solution of nonlinear systems of one dimensional balance laws abstract an asymptotic numerical method is proposed to solve nonlinear scalar dissipative balance laws as well as systems of them in one space dimension namely hyperbolic conservation laws affected by a certain kind of source term time asymptotics allows to obtain a hierarchy of coupled ordinary differential equations which can be solved by accurate methods these provide first the long time stationary solution and then corrections to it to obtain an approximation valid at lower times no accumulation of errors as time grows affects this method on the contrary results are more accurate at larger times in the scalar case an important role is played by the auxiliary function k u s u f u where f is the flux function and s is the source a similar role is played by a certain matrix in case of systems comparison is made with the godunov method and with the ahop asymptotic high order numerical methods recently developed by natalini et al

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

high order relaxation schemes for nonlinear degenerate diffusion problems several relaxation approximations to partial differential equations have been recently proposed examples include conservation laws hamilton jacobi equations convection diffusion problems and gas dynamics problems the present paper focuses on diffusive relaxation schemes for the numerical approximation of nonlinear parabolic equations these schemes are based on a suitable semilinear hyperbolic system with relaxation terms high order methods are obtained by coupling eno and weighted essentially nonoscillatory weno schemes for space discretization with implicit explicit imex schemes for time integration error estimates and a convergence analysis are developed for semidiscrete schemes with a numerical analysis for fully discrete relaxed schemes various numerical results in one and two dimensions illustrate the high accuracy and good properties of the proposed numerical schemes also in the degenerate case these schemes can be easily implemented on parallel computers and applied to more general systems of nonlinear parabolic equations in two and three dimensional cases

multistable behavior above synchronization in a locally coupled kuramoto model a system of nearest neighbors kuramoto like coupled oscillators placed in a ring is studied above the critical synchronization transition we find a richness of solutions when the coupling increases which exists only within a solvability region sr we also find that the solutions possess different characteristics depending on the section of the boundary of the sr where they appear we study the birth of these solutions and how they evolve when the coupling strength increases and determine the diagram of solutions in phase space

a neural network for a class of convex quadratic minimax problems with constraints in this paper we propose a neural network for solving a class of convex quadratic minimax problems with constraints four sufficient conditions are provided to ensure the asymptotic stability of the proposed network furthermore the exponential stability of the proposing network is also proved under certain conditions the results obtained here can be further extended to the globally projected dynamical system in addition some new stability conditions for the system are also obtained since our stability conditions can be easily checked in practice these results becomes more attractive in real applications

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

a discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws abstract we present a new relaxation approximation to scalar conservation laws in several space variables by means of semilinear hyperbolic systems of equations with a finite number of velocities under a suitable multidimensional generalization of the whitham relaxation subcharacteristic condition we show the convergence of the approximated solutions to the unique entropy solution of the equilibrium cauchy problem

large time asymptotics for partially dissipative hyperbolic systems this work is concerned with n component hyperbolic systems of balance laws in m space dimensions first we consider linear systems with constant coefficients and analyze the possible behavior of solutions as t using the fourier transform we examine the role that control theoretical tools such as the classical kalman rank condition play we build lyapunov functionals allowing us to establish explicit decay rates depending on the frequency variable in this way we extend the previous analysis by shizuta and kawashima under the so called algebraic condition sk in particular we show the existence of systems exhibiting more complex behavior than the one that the sk condition allows we also discuss links between this analysis and previous literature in the context of damped wave equations hypoellipticity and hypocoercivity to conclude we analyze the existence of global solutions around constant equilibria for nonlinear systems of balance laws our analysis of the linear case allows proving existence results in situations that the previously existing theory does not cover

global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy we consider the cauchy problem for a general one dimensional n n hyperbolic symmetrizable system of balance laws it is well known that in many physical examples for instance for the isentropic euler system with damping the dissipation due to the source term may prevent the shock formation at least for smooth and small initial data our main goal is to find a set of general and realistic sufficient conditions to guarantee the global existence of smooth solutions and possibly to investigate their asymptotic behavior for systems which are entropy dissipative a quite natural generalization of the kawashima condition for hyperbolic parabolic systems can be given in this paper we first propose a general framework for this kind of problem by using the so called entropy variables then we go on to prove some general statements about the global existence of smooth solutions under different sets of conditions in particular the present approach is suitable for dealing with most of the physical examples of systems with a relaxation extension our main tools will be some refined energy estimates and the use of a suitable version of the kawashima condition

discrete kinetic schemes for multidimensional systems of conservation laws we present here some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations which includes the relaxation schemes by s jin and z xin these schemes have a simple formulation even in the multidimensional case and do not need the solution of the local riemann problems for these approximations we give a suitable multidimensional generalization of the whitham s stability subcharacteristic condition in the scalar multidimensional case we establish the rigorous convergence of the approximated solutions to the unique entropy solution of the equilibrium cauchy problem

convergence rates for relaxation schemes approximating conservation laws in this paper we prove a global error estimate for a relaxation scheme approximating scalar conservation laws to this end we decompose the error into a relaxation error and a discretization error including an initial error omega ep we obtain the rate of convergence of sqrt ep in l1 for the relaxation step the estimate here is independent of the type of nonlinearity in the discretization step a convergence rate of sqrt del x in l1 is obtained these rates are independent of the choice of initial error omega ep thereby we obtain the order 1 2 for the total error

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

asymptotic stability of riemann solutions in bgk approximations to certain multidimensional systems of conservation laws abstract we prove the asymptotic stability of nonplanar two states riemann solutions in bgk approximations of a class of multidimensional systems of conservation laws the latter consists of systems whose flux functions in different directions share a common complete system of riemann invariants the level surfaces of which are hyperplanes the asymptotic stability to which the main result refers is in the sense of the convergence as t in l loc 1 of the space of directions x t that is the solution z t x of the perturbed cauchy problem for the corresponding bgk system satisfies x z t t d r as t in l loc 1 r n where r is the self similar entropy solution of the two states nonplanar riemann problem for the system of conservation laws

moment closure hierarchies for kinetic theories this paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory the first member of the hierarchy is the euler system which is based on maxwellian velocity distributions while the second member is based on nonisotropic gaussian velocity distributions the closure proceeds in two steps the first ensures that every member of the hierarchy is hyperbolic has an entropy and formally recovers the euler limit the second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct navier stokes behavior this is achieved through the introduction of a generalization of the bgk collision operator the simplest such system in three spatial dimensions is a 14 moment closure which also recovers the behavior of the grad 13 moment system when the velocity distributions lie near local maxwellians the closure procedure can be applied to a general class of kinetic theories

composite scheme using localized relaxation with non standard finite difference method for hyperbolic conservation laws non standard finite difference methods nsfdm introduced by mickens non standard finite difference models of differential equations world scientific singapore 1994 are interesting alternatives to the traditional finite difference and finite volume methods when applied to linear hyperbolic conservation laws these methods reproduce exact solutions in this paper the nsfdm is first extended to hyperbolic systems of conservation laws by a novel utilization of the decoupled equations using characteristic variables in the second part of this paper the nsfdm is studied for its efficacy in application to nonlinear scalar hyperbolic conservation laws the original nsfdms introduced by mickens 1994 were not in conservation form which is an important feature in capturing discontinuities at the right locations mickens construction and analysis of a non standard finite difference scheme for the burgers fisher equations journal of sound and vibration 257 4 2002 791 797 recently introduced a nsfdm in conservative form this method captures the shock waves exactly without any numerical dissipation in this paper this algorithm is tested for the case of expansion waves with sonic points and is found to generate unphysical expansion shocks as a remedy to this defect we use the strategy of composite schemes r liska b wendroff composite schemes for conservation laws siam journal of numerical analysis 35 6 1998 2250 2271 in which the accurate nsfdm is used as the basic scheme and localized relaxation nsfdm is used as the supporting scheme which acts like a filter relaxation schemes introduced by jin and xin the relaxation schemes for systems of conservation laws in arbitrary space dimensions communications in pure and applied mathematics 48 1995 235 276 are based on relaxation systems which replace the nonlinear hyperbolic conservation laws by a semi linear system with a stiff relaxation term the relaxation parameter is chosen locally on the three point stencil of grid which makes the proposed method more efficient this composite scheme overcomes the problem of unphysical expansion shocks and captures the shock waves with an accuracy better than the upwind relaxation scheme as demonstrated by the test cases together with comparisons with popular numerical methods like roe scheme and eno schemes

zero relaxation and dissipation limits for hyperbolic conservation laws we are interested in hyperbolic systems of conservation laws with relaxation and dissipation particularly the zero relaxation limit such a limit is of interest in several physical situations including gas flow near thermo equilibrium kinetic theory with small mean free path and viscoelasticity with vanishing memory in this article we study hyperbolic systems of two conservation laws with relaxation for the stable case where the equilibrium speed is subcharacteristic with respect to the frozen speeds we illustrate for a model in viscoelasticity that no oscillation develops for the nonlinear system in the zero relaxation limit for the marginally stable case where the equilibrium speed may equal one of the frozen speeds we show for a model in phase transitions that no oscillation arises when the dissipation is present and goes to zero more slowly than the relaxation our analysis includes the construction of suitable entropy pairs to derive energy estimates we need such energy estimates not only for the compactness properties but also for the deviation from the equilibrium of the solutions for the relaxation systems the theory of compensated compactness is then applied to study the oscillation in the zero relaxation limit 1993 john wiley sons inc

a discrete bgk approximation for strongly degenerate parabolic problems with boundary conditions abstract we consider a class of bgk systems with a finite number of velocities depending on a positive relaxation parameter that approximate strongly degenerate hyperbolic parabolic equations with initial boundary conditions we prove a priori estimates for the solutions of the systems showing that these functions converge towards the entropy solutions of strongly degenerate problems when the relaxation parameter goes to zero

stability of phase locking in a ring of unidirectionally coupled oscillators we discuss the dynamic behaviour of a finite group of phase oscillators unidirectionally coupled in a ring the dynamics are based on the kuramoto model in the case of identical oscillators all phase locking solutions and their stability properties are obtained for nonidentical oscillators it is proven that there exist phase locking solutions for sufficiently strong coupling an algorithm to obtain all phase locking solutions is proposed these solutions can be classified into classes each with its own stability properties the stability properties are obtained by means of a novel extension of gershgorin s theorem one class of stable solutions has the property that all phase differences between neighbouring cells are contained in contrary to intuition a second class of stable solutions is established with exactly one of the phase differences contained in the stability results are extended from sinusoidal interconnections to a class of odd functions to conclude a connection with the field of active antenna arrays is made generalizing some results earlier obtained in this field

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

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

a relaxation scheme for conservation laws with a discontinuous coefficient lwe study a relaxation scheme of the jin and xin type for conservation laws with a flux function that depends discontinuously on the spatial location through a coefficient k x if k e bv we show that the relaxation scheme produces a sequence of approximate solutions that converge to a weak solution the murat tartar compensated compactness method is used to establish convergence we present numerical experiments with the relaxation scheme and comparisons are made with a front tracking scheme based on an exact 2 x 2 riemann solver

kinetic formulation of the isentropic gas dynamics and p systems we consider the 2 x 2 hyperbolic system of isentropic gas dynamics in both eulerian or lagrangian variables also called the p system we show that they can be reformulated as a kinetic equation using an additional kinetic variable such a formulation was first obtained by the authors in the case of multidimensio nal scalar conservation laws a new phenomenon occurs here namely that the advection velocity is now a combination of the macroscopic and kinetic velocities various applications are given we recover the invariant regions deduce new l estimates using moments lemma and prove l w stability for 7 3

asymptotic preserving projective integration schemes for kinetic equations in the diffusion limit we investigate a projective integration scheme for a kinetic equation in the limit of vanishing mean free path in which the kinetic description approaches a diffusion phenomenon the scheme first takes a few small steps with a simple explicit method such as a spatial centered flux forward euler time integration and subsequently projects the results forward in time over a large time step on the diffusion time scale we show that with an appropriate choice of the inner step size the time step restriction on the outer time step is similar to the stability condition for the diffusion equation whereas the required number of inner steps does not depend on the mean free path we also provide a consistency result the presented method is asymptotic preserving in the sense that the method converges to a standard finite volume scheme for the diffusion equation in the limit of vanishing mean free path the analysis is illustrated with numerical results and we present an application to the su olson test

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

resolution d equations d evolution quasilineaires en dimension n d espace a l aide d equations lineaires en dimension n 1 on montre que l on peut calculer les solutions de l equation d evolution quasi lineaire dans r n d t u div b u a u 0 u x t r x r n t 0 et en particulier de l equation d ordre 1 d t u div b u 0 u x t r x r n t 0 a l aide d une equation d evolution lineaire dans r n r

convergence of flux vector splitting schemes with single entropy inequality for hyperbolic systems of conservation laws we prove the convergence of flux vector splitting schemes associated to hyperbolic systems of conservation laws with a single compatible entropy c we prove estimate on the l2 norm of the gradient of the numerical approximation in the inverse square root of the space increment x this estimate is related to the notion of strictly c dissipativity on f f and id f f where f f is the flux decomposition the second tool of the proof is a kinetic formulation of the flux splitting scheme with three velocities then we get a control for all entropies and apply the compensated compactness theory

a kinetic construction of global solutions of first order quasilinear equations on considere le probleme de cauchy pour des equations quasilineaires d ordre 1 de type conservation dans la region t 0 u t i 1 u a i u x i 0 u x o u o x ou x x 1 x n u u x t et a i i 1 n sont des fonctions continument differentiables a valeurs reelles d une seule variable reelle

the spectrum of the locked state for the kuramoto model of coupled oscillators we analyze the linear stability of the phase locked state in the kuramoto model of coupled oscillators the main result is the first rigorous characterization of the spectrum and its associated eigenvectors for any finite number of oscillators all but two of the eigenvalues are negative and merge into a continuous spectrum as the number of oscillators tends to infinity one eigenvalue is always zero by rotational invariance the final eigenvalue corresponding to a collective mode determines the stability of the locked state

global classical solutions for partially dissipative hyperbolic system of balance laws the basic existence theory of kato and majda enables us to obtain local in time classical solutions to generally quasilinear hyperbolic systems in the framework of sobolev spaces in x with higher regularity however it remains a challenging open problem whether classical solutions still preserve well posedness in the case of critical regularity this paper is concerned with partially dissipative hyperbolic system of balance laws under the entropy dissipative assumption we establish the local well posedness and blow up criterion of classical solutions in the framework of besov spaces with critical regularity with the aid of the standard iteration argument and friedrichs regularization method then we explore the theory of function spaces and develop an elementary fact that indicates the relation between homogeneous and inhomogeneous chemin lerner spaces mixed space time besov spaces this fact allows us to capture the dissipation rates generated from the partial dissipative source term and further obtain the global well posedness and stability by assuming at all times the shizuta kawashima algebraic condition as a direct application the corresponding well posedness and stability of classical solutions to the compressible euler equations with damping are also obtained

there is no non zero stable fixed point for dense networks in the homogeneous kuramoto model this paper is concerned with the existence of multiple stable fixed point solutions of the homogeneous kuramoto model we develop a necessary condition for the existence of stable fixed points for the general network kuramoto model this condition is applied to show that for sufficiently dense n node networks with node degrees at least 0 9395 n 1 the homogeneous equal frequencies model has only one stable fixed point solution over the full space of phase angles in the range to this is the zero fixed point solution defined by all phase angle differences being zero this result together with existing research proves a conjecture of verwoerd and mason 2007 proc of the american control conf pp 4613 8 that for the complete network and the homogeneous model the zero fixed point has a basin of attraction consisting of the entire space minus a set of measure zero the necessary conditions are also tested to see how close to sufficiency they might be by applying them to a class of regular degree networks studied by wiley et al 2006 chaos 16 015103

l1nonlinear stability of traveling waves for a hyperbolic system with relaxation where x t r r the unknowns u v belong to r the function f f u is in c r and 0 is a fixed constant relaxation systems often arise in many physical situations let us recall for example gases not in thermodynamic equilibrium vk kinetic theory ce cip pi chromatography raa river flows traffic flows and more general waves wh more recently the study of a special class of hyperbolic systems with relaxation was developed in view of the numerical approximation of discontinuous solutions of conservation laws jx an the 2 2 relaxation hyperbolic systems of conservation laws were first analyzed by liu li2 who justified some nonlinear stability criteria for diffusion waves expansion waves and traveling waves article no 0180

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

multistability of phase locking in equal frequency kuramoto models on planar graphs the number n of stable fixed points of locally coupled kuramoto models depends on the topology of the network on which the model is defined it has been shown that cycles in meshed networks play a crucial role in determining n because any two different stable fixed points differ by a collection of loop flows on those cycles since the number of different loop flows increases with the length of the cycle that carries them one expects n to be larger in meshed networks with longer cycles simultaneously the existence of more cycles in a network means more freedom to choose the location of loop flows differentiating between two stable fixed points therefore n should also be larger in networks with more cycles we derive an algebraic upper bound for the number of stable fixed points of the kuramoto model with identical frequencies under the assumption that angle differences between connected nodes do not exceed 2 we obtain n k 1c 2 int nk 4 1 which depends both on the number c of cycles and on the

entropy and global existence for hyperbolic balance laws this paper presents a general result on the existence of global smooth solutions to hyperbolic systems of balance laws in several space variables we propose an entropy dissipation condition and prove the existence of global smooth solutions under initial data close to a constant equilibrium state in addition we show that a system of balance laws satisfies a kawashima condition if and only if its first order approximation namely the hyperbolic parabolic system derived through the chapman enskog expansion satisfies the corresponding kawashima condition the result is then applied to bouchut s discrete velocity bgk models approximating hyperbolic systems of conservation laws

large time behavior of solutions for hyperbolic balance laws we study the existence and the large time behavior of global solutions to the initial value problem for hyperbolic balance laws in n space dimensions with n 3 admitting an entropy and satisfying the stable condition we first construct global existence of the solutions to such a system around a steady state if the initial energy is small enough then we show that k order derivatives of these solutions approach a constant state in the lp norm at a rate o t 12 k n2 np with p 2 and 0 n2 provided that initially z0 b 2 where b 2 is a homogeneous besov space these decay results do not impose an additional smallness assumption on lp norm of the initial data and we thus improve the results in 3 19 we also show faster decay results in the sense that if pz0 b 2 i p z0 b 2 1 with n2 n 22 k order derivatives of the solutions approach a constant state in the lp norm at a rate o t 12 k 1 n2 np

convergence to equilibrium for the relaxation approximations of conservation laws we study the cauchy problem for 2 2 semilinear and quasilinear hyperbolic systems with a singular relaxation term special comparison and compactness properties are established by assuming the subcharacteristic condition therefore we can prove the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to 0 this research was strongly motivated by the recent numerical investigations of s jin and z xin on the relaxation schemes for conservation laws 1996 john wiley sons inc