fully discrete entropy conservative schemes of arbitrary order we consider weak solutions of hyperbolic or hyperbolic elliptic systems of conservation laws in one space dimension and their approximation by finite difference schemes in conservative form the systems under consideration are endowed with an entropy entropy flux pair we introduce a general approach to construct second and third order accurate fully discrete in both space and time entropy conservative schemes in general these schemes are fully nonlinear implicit but in some important cases can be explicit or linear implicit furthermore semidiscrete entropy conservative schemes of arbitrary order are presented the entropy conservative schemes are used to construct a numerical method for the computation of weak solutions containing nonclassical regularization sensitive shock waves finally specific examples are investigated and tested numerically our approach extends the results and techniques by tadmor in numerical methods for compressible flows finite difference element and volume techniques asme new york 1986 pp 149 158 lefloch and rohde siam j numer anal 37 2000 pp 2023 2060

a level set approach to the computation of twinning and phase transition dynamics a computational method is proposed for the dynamics of solids capable of twinning and phase transitions in a two dimensional sharp interface model of twinning the stored energy function is a nonconvex potential with multiple wells the evolution of twin interfaces is governed by field equations and jump conditions of momentum balance and by a kinetic relation expressing the interface velocity as a function of the local driving traction and interfacial orientation a regularized version of the model is constructed based on the level set method a level set function which changes signs across the interface is introduced the evolution of this function is described by a hamilton jacobi equation whose velocity coefficient is determined by the kinetic relation jump conditions are thereby eliminated allowing finite difference discretization numerical simulations exhibit complex evolution of the interface including cusp formation needle growth spontaneous tip splitting and topological changes that result in microstructure refinement the results capture experimentally observed phenomena in martensitic crystals

the small dispersion limit of the korteweg de vries equation iii in parts i and ii we have derived explicit formulas for the distribution limit u of the solution of the kdv equation as the coefficient of uxxx tends to zero this formula contains n parameters 1 n whose values as well as whose number depends on x and t in section 4 we have shown that for t tb n 1 and the value of was determined in section 5 we have shown that the parameters i satisfy a nonlinear system of partial differential equations in part iii section 6 we show that for t large n 3 and we determine the asymptotic behavior of 1 2 3 and of u and u2 for t large the explicit formulas show that u and u2 are o t 1 and o t 2 respectively see formulas 6 2 and 6 24 in section 7 we study initial data whose value tends to zero as x and to 1 as x if one accepts some plausible guesses about the behavior of solutions with such initial data we derive an explicit formula for the solution and determine the large scale asymptotic behavior of the solution the function s is expressible in terms of complete elliptic integrals a similar formula is derived for u2 in section 8 we indicate how to extend the treatment of this series of papers to multihumped but still negative initial data

affordable entropy conserving and entropy stable flux functions for the ideal mhd equations in this work we design an entropy stable finite volume approximation for the ideal magnetohydrodynamics mhd equations the method is novel as we design an affordable analytical expression of the numerical interface flux function that discretely preserves the entropy of the system to guarantee the discrete conservation of entropy requires the addition of a particular source term to the ideal mhd system exact entropy conserving schemes cannot dissipate energy at shocks thus to compute accurate solutions to problems that may develop shocks we determine a dissipation term to guarantee entropy stability for the numerical scheme numerical tests are performed to demonstrate the theoretical findings of entropy conservation and robustness

why nonconservative schemes converge to wrong solutions error analysis this paper attempts to give a qualitative and quailtitative description of the numerical error introduced by using finite difference schemes in nonconservative form for scalar conservation laws we show that these schemes converge strongly in li norm to the solution of an inhomogeneous conservation law containing a borel measure source term moreover we analyze the properties of this borel measure and derive a sharp estimate for the l1 error between the limit function given by the scheme and the correct solution in general the measure source term is of the order of the entropy dissipation measure associated with the scheme in certain cases the error can be small for short times which makes it difficult to detect numerically but generically such an error will grow in time and this would lead to a large error for large time calculations finally we show that a local correction of any high order accurate scheme in nonconservative form is sufficient to ensure its convergence to the correct solution

kinetic relations and the propagation of phase boundaries in solids this paper treats the dynamics of phase transformations in elastic bars the specific issue studied is the compatibility of the field equations and jump conditions of the one dimensional theory of such bars with two additional constitutive requirements a kinetic relation controlling the rate at which the phase transition takes place and a nucleation criterion for the initiation of the phase transition a special elastic material with a piecewise linear non monotonic stress strain relation is considered and the riemann problem for this material is analyzed for a large class of initial data it is found that the kinetic relation and the nucleation criterion together single out a unique solution to this problem from among the infinitely many solutions that satisfy the entropy jump condition at all strain discontinuities

why many theories of shock waves are necessary kinetic relations for non conservative systems for a class of nonconservative hyperbolic systems of partial dif ferential equations endowed with a strictly convex mathematical entropy we formulate the initial value problem by supplementing the equations with a ki netic relation prescribing the rate of entropy dissipation across shock waves our condition can be regarded as a generalization to nonconservative systems of a similar concept introduced by abeyaratne knowles and truskinovsky for subsonic phase transitions and by lefloch for nonclassical undercompressive shocks to nonlinear hyperbolic systems the proposed kinetic relation for non conservative systems turns out to be equivalent for the class of systems under consideration at least to dal maso lefloch and murat s definition based on a prescribed family of lipschitz continuous paths in agreement with previous theories the kinetic relation should be derived from a phase plane analysis of traveling solutions associated with an augmented version of the nonconserva tive system we illustrate with several examples that nonconservative systems arising in the applications fit in our framework and for a typical model of turbulent fluid dynamics we provide a detailed analysis of the existence and properties of traveling waves which yields the corresponding kinetic function

a fully discrete scheme for diffusive dispersive conservation laws we introduce a fully discrete in both space and time scheme for the numerical approximation of diffusive dispersive hyperbolic conservation laws in one space dimension this scheme extends an approach by lefloch and rohde 4 it satisfies a cell entropy inequality and as a consequence the space integral of the entropy is a decreasing function of time this is an important stability property shared by the continuous model as well following hayes and lefloch 2 we show that the limiting solutions generated by the scheme need not coincide with the classical oleinik kruzkov entropy solutions but contain nonclassical undercompressive shock waves investigating the properties of the scheme we stress various similarities and differences between the continuous model and the discrete scheme dynamics of nonclassical shocks nucleation etc

coupling of general lagrangian systems this work is devoted to the coupling of two fluid models such a s two euler systems in lagrangian coordinates at a fixed interface we define coupling conditi ons which can be expressed in terms of continuity of some well chosen variables and then solve the coupled riemann problem in the present setting where the interface is characteristic a particula r choice of dependent variables which are transmitted ensures the overall conservativity

on the convergence of finite difference methods for pde under temporal refinement abstract in this note we present an issue in the verification of order of convergence for finite difference approximations to partial differential equations discretized in time and space when one refines both space and time together the convergence rate for a correct implementation will match that expected from an analysis of the asymptotic convergence rate or local truncation error of the method sometimes however only the time step size is reduced while the spatial mesh is held fixed the observed rate of convergence in this case may then differ from the formal order of accuracy of the method in space time in particular one class of methods second order lax wendroff time differencing produces only first order when the time step is refined at fixed spatial mesh resolution because the time and space differencing are intrinsically linked our method to arrive at this result is through analyzing the error at a defined point in time associated with many time steps we demonstrate our results computationally

the validity of the modified equation for nonlinear shock waves abstract the modified model equivalent equation is an important tool in designing and analyzing nonlinear difference schemes in this note the validity of this principle is rigorously established for nonlinear shock wave solutions and the upwind scheme in a particular case

high order schemes entropy inequalities and nonclassical shocks we are concerned with the approximation of undercompressive regularization sensitive nonclassical solutions of hyperbolic systems of conservation laws by high order accurate conservative and semidiscrete finite difference schemes nonclassical shock waves can be generated by diffusive and dispersive terms kept in balance particular attention is given here to a class of systems of conservation laws including the scalar equations and the system of nonlinear elasticity and to linear diffusion and dispersion in either the conservative or the entropy variables first we investigate the existence and the properties of entropy conservative schemes a notion due to tadmor math comp 49 1987 pp 91 103 in particular we exhibit a new five point scheme which is third order accurate at least second we study a class of entropy stable and high order accurate schemes satisfying a single cell entropy inequality they are built from any high order entropy conservative scheme by adding to it a mesh independent numerical viscosity which preserves the order of accuracy of the base scheme these schemes can only converge to solutions of the system of conservation laws satisfying the entropy inequality these entropy stable schemes exhibit mild oscillations near shocks and interestingly may converge to classical or nonclassical entropy solutions depending on the sign of their dispersion coefficient then based on a third order entropy conservative scheme we propose a general scheme for the numerical computation of nonclassical shocks importantly our scheme satisfies a cell entropy inequality following hayes and lefloch siam j numer anal 35 1998 pp 2169 2194 we determine numerically the kinetic function which uniquely characterizes the dynamics of nonclassical shocks for each regularization of the conservation laws our results compare favorably with previous analytical and numerical results finally we prove that there exists no fully discrete and entropy conservative scheme and we investigate the entropy stability of a class of fully discrete lax wendroff type schemes

the numerical viscosity of entropy stable schemes for systems of conservation laws i discrete approximations to hyperbolic systems of conservation laws are studied the amount of numerical viscosity present in such schemes is quantified and related to their entropy stability by means of comparison to this end conservative schemes which are also entropy conservative are constructed these entropy conservative schemes enjoy second order accuracy moreover they can be interpreted as piecewise linear finite element methods and hence can be formulated on various mesh configurations it is then shown that conservative schemes are entropy stable if and for three point schemes only they contain more viscosity than that present in the above mentioned entropy conservative ones

equivalent legendre polynomials numerical integration of discontinuous functions in the finite element methods abstract the efficiency of numerical integration of discontinuous functions is a challenge for many different computational methods to overcome this problem we introduce an efficient and simple numerical method for the integration of discontinuous functions on an arbitrary domain in the proposed method the discontinuous function over the domain is replaced by a continuous equivalent function using legendre polynomials the method is applicable to all ansatz spaces with continuous basis functions and it allows to use standard numerical integration methods in the entire domain this imposed continuous function is called equivalent legendre polynomial elp several numerical examples serve to demonstrate the efficiency and accuracy of the proposed method based on 2 d and 3 d problems of linear elastostatics the introduced method is further evaluated in the concept of the fictitious domain finite element methods

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

eigenvalue analysis of compatible and incompatible rectangular four node quadrilateral elements exact analytical expressions for the eigenvalues of the elastic stiffness matrix are obtained for the four node rectangular quadrilateral element a procedure is given for identifying alternative hourglass modes and eigenvalues which render the element incompatible but with non monotonic convergence assured a convergence study confirms that for the special case of when the hourglass modes coincide with beam bending the element can serve as a beam element analytical expressions are given for the resulting element stiffness matrix

a conservative and convergent scheme for undercompressive shock waves undercompressive shock waves arise in numerous physical applications we propose a class of conservative finite volume type schemes to approximate weak solutions of conservation laws that contain undercompressive shock waves we prove the convergence of a subsequence of approximate solutions towards a generalized entropy solution if the mesh width tends to zero the proof relies on a refined bv compactness analysis which accounts for the effect of the kinetic relation that drives the undercompressive wave at the same time we establish a new proof for the existence of solutions to the underlying model numerical experiments supplement the analytical results

numerical approximation of a macroscopic model of pedestrian flows this paper is concerned with the numerical approximation of the solutions of a macroscopic model for the description of the flow of pedestrians solutions of the associated riemann problem are known to be possibly nonclassical in the sense that the underlying discontinuities may well violate the lax inequalities which makes their numerical approximation very sensitive this study proposes to extend the transport equilibrium strategy proposed in c chalons c r acad sci paris ser i 342 2006 pp 623 626 and c chalons transport equilibrium schemes for computing nonclassical shocks i scalar conservation laws submitted for computing the nonclassical solutions of scalar conservation laws with either a concave convex or a convex concave flux function and supplemented with an invertible kinetic function these strong properties are not fulfilled in the present setting since the flow function admits several inflection points and the kinetic function is not invertible we nevertheless succeed in

the buckley leverett equation with dynamic capillary pressure the buckley leverett equation for two phase flow in a porous medium is modified by including dependence of the capillary pressure on the rate of change of saturation this model due to hassanizadeh and gray results in a nonlinear pseudoparabolic partial differential equation phase plane analysis including a separation function to measure the distance between invariant manifolds is used to determine when the equation supports traveling waves corresponding to undercompressive shocks the riemann problem for the underlying conservation law is solved and the structures of the various solutions are confirmed with numerical simulations of the partial differential equation

propagating phase boundaries formulation of the problem and existence via the glimm method this paper treats the hyperbolic elliptic system of two conservation laws which describes the dynamics of an elastic material having a non monotone strain stress function followingabeyaratne knowles we propose a notion of admissible weak solution for this system in the class of functions of bounded variation the formulation includes an entropy inequality a kinetic relation imposed along any subsonic phase boundary and an initiation criterion for the appearance of new phase boundaries we prove thel1 continuous dependence of the solution to the riemann problem our main result yields the existence and the stability of propagating phase boundaries the proofs are based onglimm s scheme and in particular on the techniques ofglimm andlax in order to deal with the kinetic relation we prove a result of pointwise convergence of the phase boundary

entropy stable wall boundary conditions for the three dimensional compressible navier stokes equations non linear entropy stability and a summation by parts framework are used to derive entropy stable wall boundary conditions for the three dimensional compressible navier stokes equations a semi discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator the data at the boundary are weakly imposed using a penalty flux approach and a simultaneous approximation term penalty technique although discontinuous spectral collocation operators on unstructured grids are used herein for the purpose of demonstrating their robustness and efficacy the new boundary conditions are compatible with any diagonal norm summation by parts spatial operator including finite element finite difference finite volume discontinuous galerkin and flux reconstruction correction procedure via reconstruction schemes the proposed boundary treatment is tested for three dimensional subsonic and supersonic flows the numerical computations corroborate the non linear stability entropy stability and accuracy of the boundary conditions

a well balanced and entropy conservative discontinuous galerkin spectral element method for the shallow water equations in this work we design an arbitrary high order accurate nodal discontinuous galerkin spectral element type method for the one dimensional shallow water equations the novel method uses a skew symmetric formulation of the continuous problem we prove that this discretisation exactly preserves the local mass and momentum furthermore we show that combined with a special numerical interface flux function the method exactly preserves the entropy which is also the total energy for the shallow water equations finally we prove that the surface fluxes the skew symmetric volume integrals and the source term are well balanced numerical tests are performed to demonstrate the theoretical findings

computational methods for propagating phase boundaries this paper considers numerical methods for computing propagating phase boundaries in solids described by the physical model introduced by abeyaratne and knowles the model under consideration consists of a set of conservation laws supplemented with a kinetic relation and a nucleation criterion discontinuities between two different phases are undercompressive crossing waves in the general terminology of nonstrictly hyperbolic systems of conservation laws this paper studies numerical methods designed for the computation of such crossing waves we propose a godunov type method combining front tracking with a capturing method we also consider glimm s random choice scheme both methods share the property that the phase boundaries are sharply computed in the sense that there are no numerical interior points for the description of a phase boundary this property is well known for the glimm s scheme on the other hand our front tracking algorithm is designed so that it tracks phase boundaries but captures shock waves phase boundaries are sensitive to numerical dissipation effects so the above property is essential to ensure convergence toward the correct entropy weak solution convergence of the godnuov type method is demonstrated numerically extensive numerical experiments show the practical interest of both approaches for computations of undercompressive crossing waves

second order conservative schemes and the entropy condition we consider numerical approximations to solutions of systems of hyperbolic conservation laws of the form au at af u ax 0 u e r andf r r smooth we show that conservative three point second order accurate methods cannot satisfy a local entropy inequality

non classical riemann solvers with nucleation we introduce a new nonclassical riemann solver for scalar conservation laws with concave convex flux function this solver is based on both a kinetic relation which determines the propagation speed of undercompressive nonclassical shock waves and a nucleation crite rion which makes a choice between a classical riemann solution and a nonclassical one we establish the existence of nonclassical entropy solutions of the cauchy problem and discuss several examples of wave interactions we also show the existence of a class of solutions called splitting merging solutions which are made of two large shocks and small bv bounded variation perturbations the nucleation solvers as we call them are applied to and actually motivated by the theory of thin film flows they help explain numerical results observed for such flows

error estimates for finite volume approximations of classical solutions for nonlinear systems of hyperbolic balance laws we consider a general class of finite volume schemes on unstructured but quasi uniform meshes for first order systems of hyperbolic balance laws on unstructured meshes provided the system is equipped with at least one entropy entropy flux tuple and the associated cauchy problem allows for a classical solution u we give conditions such that the finite volume approximation u h converges to u if the mesh parameter h tends to zero in fact we prove an error estimate of the form u u h l 2 le c sqrt h where c is independent of h the proof relies on a stability result for classical solutions in the class of entropy solutions due to dafermos arch rational mech anal 94 1979 pp 373 389 and diperna indiana univ math j 28 1979 pp 137 188 finally we present examples such that the conditions to apply the general convergence estimate can be satisfied at least in part the examples cover general scalar equations weakly coupled systems and the system of elastodynamics in one dimension moreover we generalize the concept of entropy conservative methods due to tadmor math comp 49 1987 pp 91 103 and show how this can be used to establish the convergence of finite volume methods for the system s case

local rotundity structure of cesaro orlicz sequence spaces abstract some criteria for extreme points and strong u points in cesaro orlicz spaces are given in consequence we find a cesaro orlicz sequence space different from c 0 which has no extreme points some examples show that in these spaces the notion of the strong u point is essentially stronger than the notion of the extreme point various examples presented in this paper show that there are some differences between criteria for extreme points and strong u points in orlicz spaces and in cesaro orlicz spaces we also show that the uniqueness of the local best approximation needs the notion of su point that is the notion of the extreme point is not strong enough here

why many theories of shock waves are necessary kinetic functions equivalent equations and fourth order models we consider several systems of nonlinear hyperbolic conservation laws describing the dynamics of nonlinear waves in presence of phase transition phenomena these models admit under compressive shock waves which are not uniquely determined by a standard entropy criterion but must be characterized by a kinetic relation building on earlier work by lefloch and collaborators we investigate the numerical approximation of these models by high order finite difference schemes and uncover several new features of the kinetic function associated with physically motivated second and third order regularization terms especially viscosity and capillarity terms on one hand the role of the equivalent equation associated with a finite difference scheme is discussed we conjecture here and demonstrate numerically that the numerical kinetic function associated with a scheme approaches the analytic kinetic function associated with the given model especially since its equivalent equation approaches the regularized model at a higher order on the other hand we demonstrate numerically that a kinetic function can be associated with the thin liquid film model and the generalized camassa holm model finally we investigate to what extent a kinetic function can be associated with the equations of van der waals fluids whose flux function admits two inflection points

nonclassical shocks and kinetic relations finite difference schemes we consider hyperbolic systems of conservation laws that are not genuinely nonlinear the solutions generated by diffusive dispersive regularizations may include nonclassical n c shock waves that do not satisfy the classical liu entropy criterion we investigate the numerical approximation of n c shocks via conservative difference schemes constrained only by a single entropy inequality the schemes are designed by comparing their equivalent equations with the continuous model and include discretizations of the diffusive and dispersive terms limits of these schemes are characterized via the kinetic relation introduced earlier by the authors we determine the kinetic function numerically for several examples of systems and schemes this study demonstrates that the kinetic relation is a suitable tool for the selection of unique n c solutions and for the study of their sensitive dependence on the critical parameters the ratios of diffusion dispersion and diffusion mesh size the shock strength and the order of discretization of the flux

stiffness matrix of the four node quadrilateral element in closed form the stiffness matrix of a plane four node quadrilateral finite element is given in closed form when expressed as a fortran subroutine and compared with the classical method of forming the stiffness matrix using gaussian integration the approach gives a cpu time speed up of the order of 2 3 on a vector machine and of the order of 4 5 on a scalar machine the technique used to generate the terms of the stiffness matrix made use of a computer algebra system which could clearly be extended to generate the matrices for other elements types

finite energy method for compressible fluids the navier stokes korteweg model this is the first of a series of papers devoted to the initial value problem for the one dimensional euler system of compressible fluids and augmented versions containing higher order terms in the present paper we are interested in dispersive shock waves and analyze the zero viscosity capillarity limit associated with the navier stokes korteweg system specifically we establish the existence of finite energy solutions as well as their convergence toward entropy solutions to the euler system our method of proof combines energy and effective energy estimates a nonlinear sobolev inequality high integrability properties for the mass density and for the velocity and compactness properties based on entropies 2015 wiley periodicals inc

a note on cesaro orlicz sequence spaces abstract the condition 2 in cesaro orlicz sequence spaces equipped with the luxemburg norm is discussed the comparison theorem for these spaces is presented some counterexamples are provided

numerical integrations of systems of conservation laws of mixed type the systems of conservation laws have been used to model dynamic phase transitions in for example the propagating phase boundaries in solids and the van der waals fluid when integrating such mixed hyperbolic elliptic systems the lax friedrichs scheme is known to give the correct solutions selected by a viscosity capillarity criterion except for a spike at the phase boundary which does not go away even with a refined mesh 15 we identify the source of this spike as an inconsistency between the lax friedrichs discretization and the viscosity capillarity equations and show a simple change of variable that can eliminate this spike we then implement a high resolution scheme for the mixed type problems that select the same viscosity capillarity solutions as the lax friedrichs scheme with higher resolutions furthermore a flexibility in the first order scheme is used to obtain solutions for a wide range of the viscosity capillarity equations

viscous shock profiles and primitive formulations weak solutions of hyperbolic systems in primitive nonconservative form for which a consistent conservation form exists are considered it is shown that for primitive formulations shock relations are not uniquely defined by the states to either side of the shock but also depend on the viscous path connecting the two consistent viscous shock profiles are enforced by adding scheme dependent small viscous perturbations that account for leading order conservation errors the resulting primitive algorithm is conservative to the order of the approximation one dimensional euler calculations of flows containing weak to moderate shocks show that conservation errors in primitive calculations are substantially reduced by including the viscous perturbation terms while not eliminating conservation errors entirely it is found that for a wide range of problems both conservative and primitive flow calculations are of comparable quality

why many theories of shock waves are necessary convergence error in formally path consistent schemes we are interested in nonlinear hyperbolic systems in nonconservative form arising in fluid dynamics and for solutions containing shock waves we investigate the convergence of finite difference schemes applied to such systems according to dal maso lefloch and murat s theory a shock wave theory for a given nonconservative system requires prescribing a priori a family of paths in the phase space in the present paper we consider schemes that are formally consistent with a given family of paths and we investigate their limiting behavior as the mesh is refined we first generalize to systems a property established earlier by hou and lefloch for scalar conservation laws and we prove that nonconservative schemes generate at the level of the limiting hyperbolic system an convergence error source term which provided the total variation of the approximations remains uniformly bounded is a locally bounded measure this convergence error measure is supported on the shock trajectories and as we demonstrate here is usually small in the special case that the scheme converges in the sense of graphs a rather strong convergence property often violated in practice then this measure source term vanishes we also discuss the role of the equivalent equation associated with a difference scheme here the distinction between scalar equations and systems appears most clearly since for systems the equivalent equation of a scheme that is formally path consistent depends upon the prescribed family of paths the core of this paper is devoted to investigate numerically the approximation of several simplified or full hyperbolic models arising in fluid dynamics this leads us to the conclusion that for systems having nonconservative products associated with linearly degenerate characteristic fields the convergence error vanishes for more general models this measure is evaluated very accurately especially by plotting the shock curves associated with each scheme under consideration as we demonstrate plotting the shock curves provide a convenient approach for evaluating the range of validity of a given scheme

a model numerical scheme for the propagation of phase transitions in solids in this paper we devise a simple finite difference scheme that produces approximations to the viscosity capillarity solutions of the equations that govern the propagation of phase transitions in solids or to the equations of van der waals fluids for all positive values of the dimensionless parameter that characterizes the viscosity capillarity solution numerical experiments showing the convergence properties of the method are presented

modelling evaporation fronts with reactive riemann solvers this work deals with the modelling of permeable fronts and the building of a numerical method allowing the multi dimensional propagation of such fronts a particular attention is given to evaporation waves that appear in cavitating systems these ones are considered as discontinuities through which a non equilibrium liquid turns to a liquid vapor mixture at thermodynamic equilibrium such transformation occurs at finite rate in order to determine this kinetics the evaporation front is assumed to propagate at the maximum admissible speed corresponding to the chapman jouguet deflagration point j r simoes moreira j e shepherd evaporation waves in superheated dodecane j fluid mech 382 1999 63 86 using this particular kinetic relation rankine hugoniot relations are closed at such fronts then it is possible to solve the associated reactive riemann problem however another difficulty is present to solve the multi dimensional propagation of permeable fronts this kind of front is subsonic and a conventional averaging scheme such as godunov scheme is inappropriate to overcome this difficulty the reactive riemann problem solution is embedded into the discrete equations method dem r abgrall r saurel discrete equations for physical and numerical compressible multiphase mixtures j comp phys 186 2003 361 396 r saurel s gavrilyuk f renaud a multiphase model with internal degrees of freedom application to shock bubble interaction j fluid mech 495 2003 283 321 this numerical method necessitates deep extensions that are detailed herein numerical results are shown and validated over experimental data some examples show that the same method may be applied to the propagation of detonation fronts

on the james constant and b convexity of cesaro and cesaro orlicz sequence spaces abstract the classical james constant and the nth james constants which are measure of b convexity for the cesaro sequence spaces ces p and the cesaro orlicz sequence spaces ces m are calculated these investigations show that ces p ces m are not uniformly non square and even they are not b convex therefore the classical cesaro sequence spaces ces p are natural examples of reflexive spaces which are not b convex moreover the james constant for the two dimensional cesaro space ces 2 2 is calculated

analytical integration formulae for linear isoparametric finite elements exact and approximate analytical expressions can be derived for integrals arising in finite element methods employing isoparametric linear quadrilaterals in two space dimensions with bilinear basis functions the formulae associated with rectangular elements arbitrarily oriented in space can be shown to be a special case the proposed method provides considerable savings in computational effort in comparison with a numerical method that employs gaussian quadrature procedures in addition the method when applied to a quadrilateral inscribable in a circle can be shown to produce better accuracy than the associated 2 2 gaussian quadrature formulae

singular limits in phase dynamics with physical viscosity and capillarity following pioneering work by fan and slemrod who studied the e ect of artificial viscosity terms we consider the sys tem of conservation laws arising in liquid vapor phase dynamics with physical viscosity and capillarity e ects taken into account following dafermos we consider self similar solutions to the rie mann problem and establish uniform total variation bounds al lowing us to deduce new existence results our analysis cover both the hyperbolic and the hyperbolic elliptic regimes and apply to arbitrarily large riemann data the proofs rely on a new technique of reduction to two coupled scalar equations associated with the two wave fans of the system strong l 1 convergence to a weak solution of bounded variation is established in the hyperbolic regime while in the hyperbolic elliptic regime a stationary singularity near the axis separating the two wave fans or more generally an almost stationary oscil lating wave pattern of thickness depending upon the capillarity viscosity ratio are observed which prevent the solution to have globally bounded variation

a numerical method for systems of conservation laws of mixed type admitting hyperbolic flux splitting the present treatment of elliptic regions via hyperbolic flux splitting and high order methods proposes a flux splitting in which the corresponding jacobians have real and positive negative eigenvalues while resembling the flux splitting used in hyperbolic systems the present generalization of such splitting to elliptic regions allows the handling of mixed type systems in a unified and heuristically stable fashion the van der waals fluid dynamics equation is used convergence with good resolution to weak solutions for various riemann problems are observed

the zero dispersion limit a deterministic analogue of turbulence weak limits of oscillatory sequences of dispersive compressible flows show a remarkable resemblance to the ensemble averages of classical turbulence theory how much relevance such ideas have for incompressible flows is not clear

a numerical study of riemann problem solutions and stability for a system of viscous conservation laws of mixed type a numerical study of the isothermal fluid equations with a nonmonotone equation of state like that of van der waals and with viscosity and capillarity terms is presented this system is ill posed i e elliptic in x vs t in some regions of state space and well posed i e hyperbolic in other regions thus it may serve as a model for describing dynamic phase transitions numerical computations of phase jumps shock waves and rarefaction waves for this system are presented although the solution of the riemann problem is not unique all of these waves are found to be stable to infinitesimal initial perturbations criteria are found for instability after o 1 initial perturbations an analytic argument is made for stability of phase transitions

on dispersive difference schemes i etude du comportement oscillatoire et de la convergence faible des solutions des approximations aux differences dispersives de u t 1 2 u 2 x 0

non classical shocks and kinetic relations scalar conservation laws this paper analyzes the non classical shock waves which arise as limits of certain diffusive dispersive approximations to hyperbolic conservation laws such shocks occur for non convex fluxes and connect regions of different convexity they have negative entropy dissipation for a single convex entropy function but not all convex entropies and do not obey the classical oleinik entropy criterion we derive necessary conditions for the existence of non classical shock waves and construct them as limits of traveling wave solutions for several diffusive dispersive approximations

high order entropy conservative schemes and kinetic relations for van der waals fluids abstract we consider the mixed hyperbolic elliptic system of two conservation laws modeling the dynamics of van der waals fluids viscosity and capillarity effects are taken into account we introduce a new class of semidiscrete high order schemes which are entropy conservative in the sense of tadmor when the viscosity is neglected and otherwise dissipate the associated mathematical entropy our numerical schemes generalize the works by e tadmor 1987 math comput 49 91 and p g lefloch and c rohde 2000 siam j numer anal 37 2023 who proposed second order and third order entropy conservative schemes respectively following b t hayes and p g lefloch 1998 siam j numer anal 35 2169 we demonstrate numerically that balanced viscosity and capillarity terms in van der waals fluids may generate nonclassical shock waves or subsonic propagating phase transitions such waves are undercompressive and do not satisfy standard entropy criteria they must be characterized by a kinetic function which we determine numerically in this paper from vanishing viscosity and capillarity the kinetic relation is an efficient tool to discuss the interplay among the viscosity capillarity and discretization parameters in van der waals fluids

on isometric copies of l and james constants in cesaro orlicz sequence spaces abstract for a wide class of orlicz functions not satisfying the growth condition 2 we show that the cesaro orlicz sequence spaces ces equipped with the luxemburg norm contain an order linearly isometric copy of l we also compute the n th james constant in these spaces for any orlicz function under either the luxemburg or orlicz norm showing that they are equal to n for any natural n 2 in particular we prove that the non trivial spaces ces are not b convex for any orlicz function

nonclassical shocks and the cauchy problem for nonconvex conservation laws abstract the riemann problem for a conservation law with a nonconvex cubic flux can be solved in a class of admissible nonclassical solutions that may violate the oleinik entropy condition but satisfy a single entropy inequality and a kinetic relation we use such a nonclassical riemann solver in a front tracking algorithm and prove that the approximate solutions remain bounded in the total variation norm the nonclassical shocks induce an increase of the total variation and therefore the classical measure of total variation must be modified accordingly we prove that the front tracking scheme converges strongly to a weak solution satisfying the entropy inequality