the relaxation time limit in the compressible euler maxwell equations abstract the aim of this paper is to study multidimensional euler maxwell equations for plasmas with short momentum relaxation time the convergence for the smooth solutions to the compressible euler maxwell equations toward the solutions to the smooth solutions to the drift diffusion equations is proved by means of the maxwell iteration as the relaxation time tends to zero meanwhile the formal derivation of the latter from the former is justified

the asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors abstract in this paper we study the asymptotic behavior of globally smooth solutions of the cauchy problem for the multidimensional isentropic hydrodynamic model for semiconductors in r d we prove that smooth solutions close to equilibrium of the problem converge to a stationary solution exponentially fast as t

the convergence of fully discrete galerkin approximations for the benjamin bona mahony bbm equation abstract in the present paper we analyze a second order in time fully discrete finite element method for the bbm equation the discretization in space is based on the standard galerkin method for the time discretization the crank nicolson scheme is used we also prove the convergence of a linearized galerkin modification scheme

error estimates of a galerkin method for some nonlinear sobolev equations in one space dimension a semidiscrete galerkin finite element method is defined and analyzed for nonlinear evolution equations of sobolev type in a single space variable optimal orderl p error estimates are derived for 2 p and it is shown that the rates of convergence of the approximate solution and its derivative are one order better than the optimal order at certain spatial jacobi and gauss points respectively also the standard nodal superconvergence results are established futher it is considered that an a posteriori procedure provides superconvergent approximations at the knots for the spatial derivatives of the exact solution

a finite element method for a version of the boussinesq equation a semidiscrete finite element method for a two point boundary value problem of a version of the boussinesq equation is analyzed the method is a combination of the h 1 galerkin method and the standard galerkin method global convergence estimates and pointwise superconvergence results are derived

on the cauchy problem for the two component camassa holm system in this paper we establish the local well posedness for the two component camassa holm system in a range of the besov spaces we also derive a wave breaking mechanism for strong solutions in addition we determine the exact blow up rate of such solutions to the system

a nonlinear pseudoparabolic equation the authors prove results on uniqueness and global existence of initial and boundary value problems for the nonlinear pseudoparabolic equation with nonhomogeneous boundary conditions a salient feature of the paper is that f and its partial derivatives are allowed to be unbounded in the special case b x t 2 a positive constant it is proved that the corresponding solution u under appropriate conditions on the data which are satisfied for example by the benjamin bona mahony equation u u 0 the solution corresponding to 0 on sufficiently small time interval a result on the asymptotic behaviour of the solution is given for t

superconvergence analysis of finite element method for the time dependent schrodinger equation in this paper we consider the two dimensional time dependent schrodinger equation firstly we use the rectangular lagrange type finite element of order p to get a semi discrete scheme of the equation and discuss the superconvergence error estimate in the h 1 norm secondly we use the crank nicolson method in time to get a fully discrete scheme of the equation and the superconvergence estimate in the h 1 norm can be obtained in this scheme finally a numerical example with the order p 1 is provided to verify our theoretical results

convergence of the euler maxwell two fluid system to compressible euler equations abstract the combined non relativistic and quasi neutral limit of two fluid euler maxwell equations for plasmas is rigorously justified in this paper for well prepared initial data the convergence of the two fluid euler maxwell system to the compressible euler equations is proved in the time interval where a smooth solution of the limit problem exists

weak galerkin finite element methods for sobolev equation we present some numerical schemes based on the weak galerkin finite element method for one class of sobolev equations in which differential operators are approximated by weak forms through the usual integration by parts in particular the numerical method allows the use of discontinuous finite element functions and arbitrary shape of element the proposed schemes will be proved to have good numerical stability and high order accuracy when time variable is continuous also an optimal error estimate is obtained for the fully discrete scheme finally some numerical results are given to verify our analysis for the scheme

finite element methods for nonlinear sobolev equations with nonlinear boundary conditions abstract we derive optimal l 2 error estimates for semi discrete finite element methods for nonlinear sobolev equations with nonlinear boundary conditions a new projection is introduced and used in the error analysis

the nonrelativistic limit in radiation hydrodynamics i weak entropy solutions for a model problem abstract this paper is concerned with a model system for radiation hydrodynamics in multiple space dimensions the system depends singularly on the light speed c and consists of a scalar nonlinear balance law coupled via an integral type source term to a family of radiation transport equations we first show existence of entropy solutions to cauchy problems of the model system in the framework of functions of bounded variation this is done by using difference schemes and discrete ordinates then we establish strong convergence of the entropy solutions indexed with c as c goes to infinity the limit function satisfies a scalar integro differential equation

geodesic flows on semidirect product lie groups geometry of singular measure valued solutions the epdiff equation or dispersionless camassa holm equation in 1d is a well known example of geodesic motion on the diff group of smooth invertible maps diffeomorphisms its recent two component extension governs geodesic motion on the semidirect product mathrm diff circleds mathcal f where mathcal f denotes the space of scalar functions this paper generalizes the second construction to consider geodesic motion on mathrm diff circleds mathfrak g where mathfrak g denotes the space of scalar functions that take values on a certain lie algebra for example mathfrak g mathcal f otimes mathfrak so 3 measure valued delta like solutions are shown to be momentum maps possessing a dual pair structure thereby extending previous results for the epdiff equation the collective hamiltonians are shown to fit into the kaluza klein theory of particles in a yang mills field and these formulations are shown to apply also at the continuum pde level in the continuum description the kaluza klein approach produces the kelvin circulation theorem

holder continuity for a modified camassa holm system this paper is concerned with the holder continuity for a modified camassa holm system we show that the date to solution for the modified camassa holm system is holder continuous from bounded set of sobolev spaces with exponent s 3 2 measured in a weaker sobolev norm with index r s on the line our result also holds for the periodic case

the fourier pseudospectral method with a restrain operator for the rlw equation in this paper we develop a new fourier pseudospectral method with a restrain operator for the modified korteweg de vries mkdv equation we prove the generalized stability of the scheme from which the convergence follows

superconvergence of a finite element approximation to the solution of a sobolev equation in a single space variable a standard galerkin method for a quasilinear equation of sobolev type using continuous piecewise polynomial spaces is presented and analyzed optimal order error estimates are established in various norms and nodal superconvergence is demonstrated discretization in time by explicit single step methods is discussed

a stabilized galerkin method for a third order evolutionary problem the periodic initial value problem for the partial differential equa tion ut uxxx s u x 2 u xx euxx outx 0 e o 0 arises in fluidization models the numerical integration of the problem is a difficult task in that many reasonable finite difference and finite element methods give rise to unstable discretizations we show how to modify the standard galerkin technique in order to stabilize it optimal order error estimates are derived and the results of numerical experiments are presented the stabilization technique suggested in the paper can be interpreted as rewriting the problem in sobolev form and would also be useful for other equations involving terms of the form u sulx

error estimates for finite difference methods for a wide angle parabolic equation we consider a model initial and boundary value problem for a third order partial differential equation pde a wide angle parabolic equation frequently used in underwater acoustics with depth and range dependent coefficients in the presence of horizontal interfaces and dissipation after commenting on the existence uniqueness theory of solution of the equation we discretize the problem by a second order finite difference method of crank nicolson type for which we prove stability and optimal order error estimates in suitable discrete l2 h1 and maximum norms we also prove under certain conditions that the forward euler scheme is also stable and convergent for the problem at hand

l2 error estimation of a quadratic finite volume element method for pseudo parabolic equations in three spatial dimensions abstract in this paper we develop a quadratic finite volume element method based on the barlow points for pseudo parabolic equations in three spatial dimensions optimal order of convergence in the l 2 norm is obtained by use of a symmetrization technique numerical results are presented to illustrate the theoretical findings

finite element galerkin method for 2d sobolev equations with burgers type nonlinearity abstract in this article the global existence of a unique strong solution to the 2d sobolev equation with burgers type nonlinearity is established using weak or weak compactness type arguments when the forcing function f 0 is in l l2 new a priori bounds are derived which are valid uniformly in time as t and with respect to the dispersion coefficient as 0 it is further shown that the solution of the sobolev equation converges to the solution of the 2d burgers equation with order o a finite element method is applied to approximate the solution in the spatial direction and the existence of a global attractor is derived for the semidiscrete scheme further using a priori bounds and an integral operator optimal error estimates are derived in l l2 norm which hold uniformly with respect to as 0 since the constants in the error estimates have exponential growth in time therefore under a certain uniqueness condition the error bounds are derived which are uniformly in time more importantly all the above results remain valid as tends to zero finally this paper concludes with some numerical examples

an asymptotic preserving scheme for the two fluid euler poisson model in the quasineutral limit this paper deals with the modeling of a plasma in the quasineutral limit using the two fluid euler poisson system in this limit explicit numerical schemes suffer from severe numerical constraints related to the small debye length and large plasma frequency here we propose an implicit scheme which reduces to a scheme for the quasineutral euler model in the quasineutral limit such a property is referred to as asymptotic preservation one of the distinctive features of this scheme is that it has a comparable numerical cost to that of an explicit scheme simply the poisson equation is replaced by a different but formally equivalent elliptic problem we present numerical simulations for two different one dimensional test cases they confirm the expected stability of the scheme in the quasineutral limit they also show that this scheme has some accuracy problems in the limit of small electron to ion mass ratio in reproducing the correct electron velocity but this problem is already present in the results of the classical algorithm numerical simulations are also performed for a two dimensional problem of a plasma expansion in vacuum between two electrodes

methods for the numerical solution of the nonlinear schroedinger equation optimal l2 rates of convergence are established for several fully discrete schemes for the numerical solution of the nonlinear schroedinger equation both finite differences and finite elements are considered for the discretization in space while the integration in time is treated either by the leap frog technique or by a modified crank nicolson procedure which generalizes the one suggested by delfour fortin and payne and possesses two useful conserved quantities

rigorous derivation of incompressible type euler equations from non isentropic euler maxwell equations abstract in this paper we investigate the convergence of the time dependent and non isentropic euler maxwell equations to incompressible euler equations in a torus via the combined quasi neutral and non relativistic limit for well prepared initial data the convergences of solutions of the former to the solutions of the latter are justified rigorously by an analysis of asymptotic expansions and energy method

some properties of solutions to the periodic two component hyperelastic rod wave equation abstract in this paper we study the cauchy problem of the periodic two component hyperelastic rod wave equation we first establish the local well posedness and precise blow up scenario then we obtain several blow up results and the blowup rate of strong solutions moreover we present two global existence results for strong solutions finally we consider analyticity of solutions and the initial boundary value problem

global solutions to a special case of the generalized weakly dissipative periodic two component camassa holm system abstract we establish global well posedness to the generalized weakly dissipative periodic two component camassa holm system in some special case

well posedness and analyticity for the cauchy problem for the generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem for the generalized camassa holm equation which was proposed by hakkaev and kirchev s hakkaev k kirchev local well posedness and orbital stability of solitary wave solutions for the generalized camassa holm equation comm partial differential equations 30 2005 761 781 we establish the local well posedness for a range of besov spaces moreover with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time which extends some results of danchin r danchin a few remarks on the camassa holm equation differential integral equations 14 2001 953 988 and himonas and misiolek a himonas g misiolek analyticity of the cauchy problem for an integrable evolution equation math ann 327 2003 575 584 to more general equations

the diffusive relaxation limit of non isentropic euler maxwell equations for plasmas abstract this paper concerns the non isentropic euler maxwell equations for plasmas with short momentum relaxation time with the help of the maxwell type iteration it is obtained that as the relaxation time tends to zero periodic initial value problem of certain scaled non isentropic euler maxwell equations has unique smooth solutions existing in the time interval where the corresponding classical drift diffusion model has smooth solutions meanwhile we justify a formal derivation of the corresponding drift diffusion model from the non isentropic euler maxwell equations

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

b spline quasi interpolation based numerical methods for some sobolev type equations in this article we intend to use quadratic and cubic b spline quasi interpolants to develop higher order numerical methods for some sobolev type equations in one space dimension our aim is also to compare the performance of the proposed methods in terms of the accuracy and the rate of convergence we also discuss another approach to the cubic b spline quasi interpolation based method where we achieve fourth order of accuracy in space we theoretically establish the order of accuracy for the three proposed methods and also establish the l 2 stability in the linear case using von neumann analysis as a particular case of the sobolev type equations we take the equal width and the benjamin bona mahony burgers equations and perform several numerical experiments to support our theoretical results

fourier spectral methods for pseudo parabolic equations fourier spectral methods of galerkin and collocation type for quasilinear pseudo parabolic equations are presented convergence with spectral accuracy is proved for both galerkin and collocation approximations unconditionally stable discretization in time by explicit one step methods is analyzed fast transform methods to compute the nonlinear terms are proposed

singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite using classical methods the authors develop a general theory of such problems this theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto fluid dynamics including new constructive local existence theorems for the time singular limit equations in particular the authors give an entirely self contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the mach number becomes small the theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures similar results are developed also for the compressible and incompressible navier stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero

on the convergence of difference schemes for the benjamin bona mahony bbm equation abstract in this paper we analyze a crank nicolson type finite difference scheme for the bbm equation we prove the second order convergence in discrete h 1 norm of the difference approximate solutions the existence stability and uniqueness are also discussed in detail at last a linearized difference scheme is given and its convergence is also proved

on a camassa holm type equation with two dependent variables we consider a generalization of the camassa holm ch equation with two dependent variables called ch2 introduced in a paper by liu and zhang liu s q and zhang y 2005 j geom phys 54 427 53 we briefly provide an alternative derivation of it based on the theory of hamiltonian structures on the dual of a lie algebra the lie algebra involved here is the same algebra as underlies the nls hierarchy we study the structural properties of the hierarchy defined by the ch2 equation within the bi hamiltonian theory of integrable pdes and provide its lax representation then we explicitly discuss how to construct classes of solutions both of peakon and of algebro geometrical type finally we sketch the construction of a class of singular solutions defined by setting to zero one of the two dependent variables

numerical schemes for a model for nonlinear dispersive waves abstract a description is given of a number of numerical schemes to solve an evolution equation that arises when modelling the propagation of water waves in a channel the discussion also includes the results of numerical experiments made with each of the schemes it is suggested on the basis of these experiments that one of the schemes may have discrete solitary wave solutions

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

equivalent formulations and numerical schemes for a class of pseudo parabolic equations this paper investigates three different formulations for a class of pseudo parabolic equations such equations are encountered for example as a model for two phase porous media flows when dynamic effects in the capillary pressure are included we first show the equivalence of the three different formulations for the original equation on the basis of this we further investigate the corresponding discretization in time and give some numerical examples

numerical schemes for a pseudo parabolic burgers equation discontinuous data and long time behaviour we consider a simplified model for vertical non stationary groundwater flow which includes dynamic capillary pressure effects specifically we consider a viscous burgers type equation that is extended with a third order term containing mixed derivatives in space and time we analyse the one dimensional boundary value problem and investigate numerically its long time behaviour the numerical schemes discussed here take into account possible discontinuities of the solution

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

a superconvergent finite element method for the korteweg de vries equation an unconditionally stable fully discrete finite element method for the korteweg de vries equation is presented in addition to satisfying optimal order global estimates it is shown that this method is superconvergent at the nodes the algorithm is derived from the conservative method proposed by the second author by the introduction of a small time independent forcing term into the discrete equations this term is a form of the quasiprojection which was first employed in the analysis of superconvergence phenomena for parabolic problems however in the present work unlike in the parabolic case the quasiprojection is used as perturbation of the discrete equations and does not affect the choice of initial values

the non relativistic limit of euler maxwell equations for two fluid plasma abstract this paper is concerned with two fluid time dependent non isentropic euler maxwell equations in a torus for plasmas or semiconductors by using the method of formal asymptotic expansions we analyze the non relativistic limit for periodic problems with the prepared initial data it is shown that the small parameter problems have unique solutions existing in the finite time interval where the corresponding limit problems compressible euler poisson equations have smooth solutions moreover the formal limit is rigorously justified by an iterative scheme and an analysis of asymptotic expansions up to any order

convergence of the nonisentropic euler maxwell equations to compressible euler poisson equations this paper is concerned with the convergence of the time dependent and nonisentropic euler maxwell equations to compressible euler poisson equations in a torus via the nonrelativistic limit by using the method of formal asymptotic expansions we analyze the nonrelativistic limit for periodic problems with the prepared initial data it is shown that the small parameter problem has unique solutions existing in the finite time interval where the corresponding limit problems have smooth solutions furthermore the convergence of solutions is rigorously justified

averaging for superconvergence verification and application of 2d edge elements to maxwell s equations in metamaterials abstract in this paper a simple averaging technique for edge elements is justified to improve the standard convergence rate hence achieving superconvergence when they are used to solve the maxwell s equations for simplicity here we focus on the lowest order triangular edge element which is widely used in practice though there exists no natural superconvergence points for the numerical solution obtained on such an edge element superconvergence can be obtained after a simple average of solutions over the neighboring elements a comprehensive analysis for the lowest order triangular edge element is carried out and one order higher convergence rate than the standard interpolation error estimate is proved for the averaged solution at midpoints of those interior edges of parallelograms formed by two triangles i e superconvergence happens at the parallelogram centers we also provide detailed analysis to explain why several cases do not have superconvergence extensive numerical results consistent with our analysis are presented

analysis of second order finite volume element methods for pseudo parabolic equations in three spatial dimensions abstract in this paper we study a finite volume element approximation of pseudo parabolic equations in three spatial dimensions the triquadratic elements are used as solution space to obtain a high order convergence rate based on some special dual partition we prove the optimal error estimate in h 1 norm of second order accuracy numerical results are also given at the end to demonstrate the efficiency of the method