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

nonclassical interactions portrait in a macroscopic pedestrian flow model abstract in this paper we describe the main characteristics of the macroscopic model for pedestrian flows introduced in r m colombo m d rosini pedestrian flows and non classical shocks math methods appl sci 28 13 2005 1553 1567 and recently sperimentally verified in d helbing a johansson h z al abideen dynamics of crowd disasters an empirical study phys rev e statistical nonlinear and soft matter physics 75 4 2007 046109 after a detailed study of all the possible wave interactions we prove the existence of a weighted total variation that does not increase after any interaction this is the main ingredient used in r m colombo m d rosini existence of nonclassical cauchy problem modeling pedestrian flows technical report brescia department of mathematics 2008 to tackle the cauchy problem through wave front tracking see a bressan hyperbolic systems of conservation laws the one dimensional cauchy problem oxford lecture ser math appl vol 20 oxford univ press oxford 2000 the one dimensional cauchy problem a bressan the front tracking method for systems of conservation laws in c m dafermos e feireisl eds handbook of differential equations evolutionary equations vol 1 elsevier 2004 pp 87 168 r m colombo wave front tracking in systems of conservation laws appl math 49 6 2004 501 537 from the mathematical point of view this model is one of the few examples of conservation laws in which nonclassical solutions have a physical motivation see p g lefloch hyperbolic systems of conservation laws lectures math eth zurich birkhauser basel 2002 the theory of classical and nonclassical shock waves and an existence result is available

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

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

on solitary wave solutions of a class of nonlinear partial differential equations based on the function 1 coshn αx βt abstract the method of simplest equation is applied for obtaining exact solitary traveling wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives the used simplest equation is f 2 n 2 f 2 f 2 n 2 n the developed methodology is illustrated on examples of classes of nonlinear partial differential equations that contain i only monomials of odd grade with respect to participating derivatives ii only monomials of even grade with respect to participating derivatives iii monomials of odd and monomials of even grades with respect to participating derivatives the obtained solitary wave solution for the case i contains as particular cases the solitary wave solutions of korteweg devries equation and of a version of the modified korteweg devries equation one of the obtained solitary wave solutions for the case ii is a solitary wave solution of the classic version of the boussinesq type equation

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

pedestrian movement analysis in transfer station corridor velocity based and acceleration based abstract in this paper pedestrians are classified into aggressive and conservative ones by their temper aggressive pedestrians walking through crowd in transfer station corridor is analyzed treating pedestrians as particles this paper uses the modified social force model msfm as the building block where forces involve self driving force repulsive force and friction force the proposed model in this paper is a discrete model combining the msfm and cellular automata ca model where the updating rules of the ca are redefined with msfm due to the continuity of values generated by the msfm we use the fuzzy logic to discretize the continuous values into cells pedestrians can move in one step with the observation that stimulus around pedestrians influences their acceleration directly an acceleration based movement model is presented compared to the generally reviewed velocity based movement model in the acceleration based model a discretized version of kinematic equation is presented based on the acceleration discretized with fuzzy logic in real life some pedestrians would rather keep their desired speed and this is also mimicked in this paper which is called inertia compared to the simple triangular membership function a trapezoidal membership function and a piecewise linear membership function are used to capture pedestrians inertia with the trapezoidal and the piecewise linear membership function many overlapping scenarios should be carefully handled and dubois and prade s four index method is used to completely describe the relative relationship of fuzzy quantities finally a simulation is constructed to demonstrate the effect of our model

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

isogeometric analysis of the isothermal navier stokes korteweg equations this paper is devoted to the numerical simulation of the navier stokes korteweg equations a phase eld model for water water vapor two phase ows we develop a numerical formulation based on isogeometric analysis that permits straightforward treatment of the higher order partial dierenti al operator that represents capillarity we introduce a new renement methodology that desensitizes the numerical solution to the computational mesh and achieves mesh invariant solutions finally we present several numerical examples in two and three dimensions that illustrate the eectiveness and robustness of our approach

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

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

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

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

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

on the general linear coupled system for diffusion in media with two diffusivities abstract for the general linear coupled system of partial differential equations arising in the theory of diffusion in media with double diffusivity simple uniqueness criteria and a method of solution of boundary value problems are established the equations studied retain the so called cross terms which have been neglected in all previous investigations moreover these equations arise as generalizations of a number of existing theories for example heat flow in heterogeneous multicomponent systems flow of water in fissured rocks and a model of an arms race the simple inequalities obtained on the various constants of the theory which guarantee uniqueness of solutions and existence of source solutions might serve as guidelines in an experimental determination of these constants the solution procedure involves solving two boundary value problems for the classical diffusion equation and the formulae given mean that closed form expressions can be deduced for a number of commonly occurring boundary value problems the paper emphasizes the general equations without special reference to particular physical applications or boundary value problems

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

modified method of simplest equation and its applications to the bogoyavlenskii equation in this paper we construct the exact traveling wave solutions of the bogoyavlenskii equation using modified method of simplest equation the simplest equation herewith is a special riccati equation in the meantime the proposed method in this work is proved to be a powerful mathematical tool for obtaining exact solutions of other nonlinear partial differential equations in mathematical physics and other research fields

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

extended simplest equation method for nonlinear differential equations abstract the modified simplest equation method to look for the exact solutions of the nonlinear differential equations is presented our approach is applied to search for the exact solutions of the sharma tasso olver and the burgers huxley equations the new exact solutions of these equations are obtained

the uniqueness and stability of the solution of the riemann problem of a system of conservation laws of mixed type we establish the uniqueness and stability of the similarity solution of the riemann problem for a 2 2 system of conservation laws of mixed type with initial data separated by the elliptic region which satisfies the viscosity capillarity travelling wave admissibility criterion

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

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

solving coupled systems of linear second order differential equations knowing a part of the spectrum of the companion matrix abstract in this paper an explicit closed form solution of initial value problems for coupled systems of time invariant second order differential equations is given without computing the exponential of the associated companion matrix

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

existence of solitary solutions in a class of nonlinear differential equations with polynomial nonlinearity the inverse balancing method for the determination of the necessary conditions of existence of solitary solutions to mth order differential equations with nth order polynomial nonlinearity is presented in this paper it is shown that the order of possible solitary solutions does not increase if orders of the differential equation and the polynomial nonlinearity increase furthermore the relationships between the order of the solitary solution and the order of the equation and the nonlinearity are given in the explicit form

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

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

diffusive dispersive traveling waves and kinetic relations part i nonconvex hyperbolic conservation laws abstract motivated by the theory of phase transition dynamics we consider one dimensional nonlinear hyperbolic conservation laws with nonconvex flux function containing vanishing nonlinear diffusive dispersive terms searching for traveling wave solutions we establish general results of existence uniqueness monotonicity and asymptotic behavior in particular we investigate the properties of the traveling waves in the limits of dominant diffusion dominant dispersion and asymptotically small or large shock strength as the diffusion and dispersion parameters tend to 0 the traveling waves converge to shock wave solutions of the conservation law which either satisfy the classical oleinik entropy criterion or are nonclassical undercompressive shocks violating it

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

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

a vanishing viscosity approach on the dynamics of phase transitions in van der waals fluids abstract the existence of solutions for the riemann problem of a mixed type system is established by a vanishing similarity viscosity approach the solutions that we construct are also admissible by the travelling wave criterion derived from the most common form of viscosity the assumption needed includes the constitutive function of van der waals fluids

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

solitary wave solutions for nonlinear partial differential equations that contain monomials of odd and even grades with respect to participating derivatives the method of simplest equation is applied to nonlinear pdes the pdes contain monomials of odd and even order with respect to derivatives solitary wave solutions are studied numerous examples of the methodology are presented we apply the method of simplest equation for obtaining exact solitary traveling wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives we consider first the general case of presence of monomials of the both odd and even grades and then turn to the two particular cases of nonlinear equations that contain only monomials of odd grade or only monomials of even grade the methodology is illustrated by numerous examples

on some solutions of the two dimensional sine gordon equation there exists an approach for finding of exact solutions of the two dimensional sine gordon equation using this approach three classes of solutions have been found one of the classes consists of running wave solutions which are a generalization of the solutions of the one dimensional sine gordon equation this class of solutions is studied here

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

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

running wave solutions of the two dimensional sine gordon equation after the lamb substitution the 2d sine gordon equation was solved three classes of solutions for this equation were found ad hoc the 2d sine gordon equation was reduced to an algebraic system consisting of three equations the solutions given of the 2d sine gordon equation are a generalization of the solutions of the 1d sine gordon equation they are also a generalization of the solutions of the equation phi xx yy sin phi x y

new class of running wave solutions of the 2 1 dimensional sine gordon equation a new class of running wave solutions of the 2 1 dimensional sine gordon equation is investigated the obtained waves require two spatial dimensions for their propagation i e they generalize solutions of the 2 0 dimensional sine gordon equation the parameters of the waves strongly depend on the wave amplitude and there exist forbidden areas for the wavenumber and frequency the obtained solutions describe a new class of josephson waves whose velocity is smaller than the swihart velocity if omega 0 the running waves are reduced to the self consistent phase current and magnetic field distributions in a large two dimensional josephson junction the self restriction coefficient for the josephson current corresponding to one of the structures is calculated

modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations we discuss the application of a variant of the method of simplest equation for obtaining exact traveling wave solutions of a class of nonlinear partial differential equations containing polynomial nonlinearities as simplest equation we use differential equation for a special function that contains as particular cases trigonometric and hyperbolic functions as well as the elliptic function of weierstrass and jacobi we show that for this case the studied class of nonlinear partial differential equations can be reduced to a system of two equations containing polynomials of the unknown functions this system may be further reduced to a system of nonlinear algebraic equations for the parameters of the solved equation and parameters of the solution any nontrivial solution of the last system leads to a traveling wave solution of the solved nonlinear partial differential equation the methodology is illustrated by obtaining solitary wave solutions for the generalized korteweg devries equation and by obtaining solutions of the higher order korteweg devries equation

nonlinear differential equations of the second third and fourth order with exact solutions abstract we present the class of nonlinear ordinary differential equations of the second third and fourth order in a polynomial form with exact solutions this class has exact solutions in the form of solitary waves kinks and one periodic solutions

on nonlinear population waves we discuss a model system of partial differential equations for description of the spatio temporal dynamics of interacting populations we are interested in the waves caused by migration of the populations we assume that the migration is a diffusion process influenced by the changing values of the birth rates and coefficients of interaction among the populations for the particular case of one population and one spatial dimension the general model is reduced to analytically tractable pde with polynomial nonlinearity up to 4th order we investigate this particular case and obtain two kinds of solutions i approximate solution for small value of the ratio between the coefficient of diffusion and the wave velocity and ii exact solutions which describe nonlinear kink and solitary waves in an appropriate phase space the kinks correspond to a connection between two states represented by a saddle point and a stable node finally we derive conditions for the asymptotic stability of the obtained solutions

an initial boundary value problem of physiological significance for equations of nerve conduction a model of nerve conduction that has gained wide acceptance in the biophysical community is that for the giant axon of the squid loligo of hodgkin and huxley 5 this nonlinear parabolic partial differential equation pde when considered as part of a particular initial boundary value problem ibvp models the electrical activity of an axon under conditions found both in nature and in the laboratory this ibvp for the hodgkin huxley equations is proved to be well posed in the sense of hadamard 3 and a priori bounds on the solution are derived

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

a limiting viscosity approach to the riemann problem for materials exhibiting a change of phase ii describes the one dimensional isothermal motion of a compressible elastic fluid or solid in lagrangian coordinates considerable study has been made for the case p w 0 in which the system 1 i a b is hyperbolic however in some material models for instance in the van der waals fluid p w may be positive for some values of w in this paper we consider a function p w with graph shown in fig 1 more precisely we assume that p e c1 and

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

traveling waves and statistical distributions connected to systems of interacting populations we discuss the following two issues from the dynamics of interacting populations i density waves for the case or negligible random fluctuations of the population densities ii probability distributions connected to the model equations for spatially averaged population densities for the case of significant random fluctuations of the independent quantity that can be associated with the population density for the case of issue i we consider model equations containing polynomial nonlinearities such nonlinearities arise as a consequence of interaction among the populations for the case of large population densities or as a result of a taylor series expansion for the case of small density of interacting populations by means of the modified method of the simplest equation we obtain exact traveling wave solutions of the model equations and these solution for the case of issue ii we discuss model equations of the fokker planck kind for the evolution of the statistical distributions of population densities we derive a few stationary distributions for the population density and calculate the expected exit time associated with the extinction of the studied population

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

a random sampling method for a family of temple class systems of conservation laws the aw rascle zhang traffic model a model of sedimentation and other applications lead to nonlinear 2 times 2 systems of conservation laws that are governed by a single scalar system velocity such systems are of the temple class since rarefaction wave curves and hugoniot curves coincide moreover one characteristic field is genuinely nonlinear almost everywhere and the other is linearly degenerate two well known problems associated with these systems are handled via a random sampling approach firstly godunov s and related methods produce spurious oscillations near contact discontinuities since the numerical solution invariably leaves the invariant region of the exact solution it is shown that alternating between averaging av and remap steps similar to the approach by chalons and goatin commun math sci 5 533 551 2007 generates numerical solutions that do satisfy an invariant region property if the remap step is made by random sampling rs then combined techniques due to glimm commun pure appl math 18 697 715 1965 leveque and temple trans am math soc 288 115 123 1985 prove that the resulting av rs scheme converges to a weak solution numerical examples illustrate that the new scheme is superior to godunov s method in accuracy and resolution secondly the vacuum state which may form even from positive initial data causes potential problems of non uniqueness and instability this is resolved by introducing an alternative riemann solution concept

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

application of the method of simplest equation for obtaining exact traveling wave solutions for the extended korteweg de vries equation and generalized camassa holm equation a version of the method of the simplest equation called modified method of simplest equation is applied to the extended korteweg de vries equation and to generalized camassa holm equation exact traveling wave solutions of these two nonlinear partial differential equations are obtained the equations of bernoulli riccati and the extended tanh equation are used as simplest equations some of the obtained solutions correspond to surface water waves

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

modified method of simplest equation and its application to nonlinear pdes abstract we search for traveling wave solutions of the class of pdes p 1 n 1 a p q p q t p r 2 n 2 b r q q t r s 1 n 3 c s q s q x s u 2 n 4 d u q q x u f q 0 where a p q b r q c s q d u q and f q are polynomials of q the basis of the investigation is a modification of the method of simplest equation the equations of bernoulli riccati and the extended tanh function equation are used as simplest equations the obtained general results are illustrated by obtaining exact solutions of versions of the generalized kuramoto sivashinsky equation reaction diffusion equation with density dependent diffusion and the reaction telegraph equation

analytic solution with a priori error bounds for a class of mixed coupled partial differential equations in this paper the existence of a sequence of stable solutions xn of the parametric matrix equation z2 az n2b 0 so that xnn converges to a stable matrix is studied the result is used for solving mixed problems related to the coupled second order partial differential system utt auxx but 0

diffusive dispersive travelling waves and kinetic relations v singular diffusion and nonlinear dispersion we consider scalar hyperbolic conservation laws with non convex flux and vanishing nonlinear and possibly singular diffusion and dispersion terms the diffusion has the form r u u x x and we cover for instance the singular diffusion u x p u x x where p 0 is arbitrary we investigate the existence uniqueness and various properties of classical and non classical travelling waves and of the kinetic function the latter serves to characterize non classical shock waves via an additional algebraic constrain called a kinetic relation we discover that p is a somewhat unexpected critical value for p we obtain properties that are qualitatively similar to those we established earlier for regular and linear diffusion however for p the behaviour of the kinetic function is very different as for instance non classical shocks can have arbitrary small strength the behaviour of the kinetic function near the origin is carefully investigated and depends on whether p p or p in particular in the special case of the cubic flux function and for the regularization u x p u x x with p 0 or 1 the kinetic function can be computed explicitly when p the kinetic function is simply a linear function of its argument

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