time behaviour of the error when simulating finite band periodic waves the case of the kdv equation this paper is devoted to study the error growth of numerical time integrators for n phase or n band quasi periodic in time solutions of the periodic korteweg de vries equation it is shown that the preservation through numerical time integration of conserved quantities of the periodic problem of the equation may be an element to take into account in the selection of the numerical method we explain why the inclusion of these properties of conservation provides a better error propagation in particular we emphasize how the preservation of invariants makes influence in the simulation of some physical parameters of the waves

backward error analysis for multi symplectic integration methods a useful method for understanding discretization error in the numerical solution of odes is to compare the system of odes with the modified equations obtained through backward error analysis and using symplectic integration for hamiltonian odes provides more incite into the modified equations in this paper the ideas of symplectic integration are extended to hamiltonian pdes and this paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these types of discretizations in particular local conservation laws of energy and momentum are not preserved exactly when symplectic integrators are used to discretize but the modified equations are used to derive modified conservation laws that are preserved to higher order along the numerical solution these results are also applied to the nonlinear wave equation

a meshless symplectic method for two dimensional schrdinger equation with radial basis functions in this study we propose a meshless symplectic algorithm for two dimensional time dependent schrdinger equation with radial basis functions rbfs approximation theory the truncation error and global error are also investigated the numerical experiments show that the scheme is higher order accuracy easy to implement with the scattered knots furthermore the algorithm possesses a long time tracking capability

error propagation when approximating multi solitons the kdv equation as a case study abstract the paper studies the influence of the time discretizations when simulating some phenomena involving more than one solitary wave taking the kdv equation as a case study we obtain some conditions on the numerical method in order to get a more correct simulation of multi soliton solutions they are related to the evolution of the conserved quantities of the problem through the numerical integration it is shown that when approximating n solitons a method that preserves n invariants of the problem shows a better time propagation of the error than that of a general scheme as a consequence of this the simulation of some physical parameters that characterize the waves is more suitable when using conservative integrators we also show how these results can be extended to the approximation to multi solitons of any equation of the kdv hierarchy and more generally other integrable equations

backward error analysis for multistep methods in recent years much insight into the numerical solution of ordinary differential equations by one step methods has been obtained with a backward error analysis it allows one to explain interesting phenomena such as the almost conservation of energy the linear error growth in hamiltonian systems and the existence of periodic solutions and invariant tori in the present article the formal backward error analysis as well as rigorous exponentially small error estimates are extended to multistep methods a further extension to partitioned multistep methods is outlined and numerical illustrations of the theoretical results are presented

error growth in the numerical integration of periodic orbits with application to hamiltonian and reversible systems we analyze in detail the growth with time of the coefficients of the asymptotic expansion of the error in the numerical integration with one step methods of periodic solutions of systems of ordinary differential equations variable stepsizes are allowed we successively consider general hamiltonian and reversible problems for hamiltonian and reversible systems and under fairly general hypotheses on the orbit being integrated numerical methods with relevant geometric properties symplecticness energy conservation reversibility are proved to have better error growth than general methods

symplectic integrator for nonlinear high order schrodinger equation with a trapped term in the manuscript we discuss the symplectic integrator for the numerical solution of a kind of high order schrodinger equation with trapped terms the hamiltonian formulism is discovered for it we first discretize the hamiltonian system in space to reduce it to a finite dimensional one then the symplectic midpoint scheme is applied to the temporal discretization the symplectic scheme we devise is of second order accuracy in time and 2lth order accuracy in space it is proved that it preserves the charge of the original equation veraciously the energy is not preserved explicitly for the exception of the linear case however after some computation the energy transit formula in the temporal direction is obtained in the numerical part we compare our scheme with some existing schemes including the leap frog scheme and the energy preserving scheme from the numerical evidence we find that our numerical schemes are efficient and available the numerical results are consistent with the theoretical analysis

new variational and multisymplectic formulations of the euler poincare equation on the virasoro bott group using the inverse map we derive a new variational principle leading to a new momentum map and a new multisymplectic formulation for a family of euler poincare equations defined on the virasoro bott group by using the

dispersive properties of multisymplectic integrators multisymplectic ms integrators i e numerical schemes which exactly preserve a discrete space time symplectic structure are a new class of structure preserving algorithms for solving hamiltonian pdes in this paper we examine the dispersive properties of ms integrators for the linear wave and sine gordon equations in particular a leapfrog in space and time scheme a member of the lobatto runge kutta family of methods and the preissman box scheme are considered we find the numerical dispersion relations are monotonic and that the sign of the group velocity is preserved the group velocity dispersion gvd is found to provide significant information and succinctly explain the qualitative differences in the numerical solutions obtained with the different schemes further the numerical dispersion relations for the linearized sine gordon equation provides information on the ability of the ms integrators to capture the sine gordon dynamics we are able to link the numerical dispersion relations to the total energy of the various methods thus providing information on the coarse grid behavior of ms integrators in the nonlinear regime

a meshless symplectic algorithm for nonlinear wave equation using highly accurate rbfs quasi interpolation this study suggests a high order meshless symplecitc algorithm for hamiltonian wave equation by using highly accurate radial basis functions rbfs quasi interpolation operator the method does not require solving a resultant full matrix and possesses a high order accuracy compared with existing numerical methods we also present a theoretical framework to show the conservativeness and convergence of the proposed symplectic method as the numerical experiments shown it not only offers a high order accuracy but also has a good property of long time tracking capability

cubic forms in 14 variables the result can be rephrased in geometric language to say that any projective cubic hypersurface defined over q of dimension at least 12 has a q point davenport s result was extended to arbitrary number fields by pleasants 9 and it would be interesting to know whether theorem 1 could similarly be extended these results can be seen as an attempt to extend the classical theorem of meyer 1884 from quadratic forms to cubic forms meyer showed that any indefinite quadratic from over z in 5 or more variables must represent zero indeed meyer s result was generalized by minkowski who showed that a quadratic form over z in any number of variables represents zero if and only if it represents zero over every completion of q it is a well known fact that this local condition is automatically satisfied for qp as soon as n 5 thus meyer s result requires only the condition of indefiniteness the analogous fact for cubic forms is that p adic zeros exist whenever n 10 see davenport 6 chapter 18 for example of course the condition for r holds for any n in this case thus it is natural to conjecture that theorem 1 should hold as soon as n 10 indeed for smaller values of n one might expect that it should suffice for the form c x to represent zero in each field qp however when n 3 or 4 it is possible for a cubic form to have zeros in every completion of q without there being a global zero this is shown by the examples

geometric integrators for the nonlinear schrodinger equation abstract recently an interesting new class of pde integrators multisymplectic schemes has been introduced for solving systems possessing a certain multisymplectic structure some of the characteristic features of the method are its local nature independent of boundary conditions and an equal treatment of spatial and temporal variables the nonlinear schrodinger equation nls has a multisymplectic formulation and in this paper we discuss the performance of both symplectic and multisymplectic integrators for the nls in the numerical experiments the multisymplectic concatenated midpoint scheme a centered cell discretization is shown to preserve the local conservation laws extremely well over long times and to preserve global invariants such as the norm and momentum within roundoff on the other hand an integrable hamiltonian semi discretization of nls from ablowitz and ladik al possesses a full set of global conservation laws and a noncanonical symplectic structure we generalize the generating function technique to develop symplectic integrators of arbitrary order for a general class of noncanonical systems carrying a symplectic structure of the al type another approach examined in the paper is the introduction of transformations to reduce the al system to either 1 separable form or 2 canonical form and then apply standard schemes in the new coordinates all of the discretizations are tested numerically using initial data for spatially periodic multiphase solutions the performance of the schemes as well as interrelations among various geometric features are discussed

a robust numerical integrator for the short pulse equation near criticality abstract the short pulse equation was introduced by schafer wayne 2004 for modeling the propagation of ultrashort optical pulses while it can describe a wide range of solutions its ultrashort pulse solutions with a few cycles which the conventional nonlinear schrodinger equation does not possess have drawn much attention in such a region the solutions can become quite close to a singularity and thus existing numerical methods cease to work stably or even if they do they require high computational cost in this paper we propose a robust numerical integration method which is obtained by combining a hodograph transformation and some structure preserving methods the resulting scheme successfully works even when the singularity occurs and thus gives a highly robust method for resolving near singular solutions of interest it is confirmed by numerical experiments and some new insights about derivative blow up are obtained

unstable eigenvalues and the linearization about solitary waves and fronts with symmetry the linear stability of solitary wave or front solutions of hamiltonian evolutionary equations which are equivariant with respect to a lie group is studied the organizing centre for the analysis is a multisymplectic formulation of hamiltonian pdes where a distinct symplectic operator is assigned for time and space this separation of symplectic structures leads to new characterizations of the following components of the analysis the states at infinity are characterized as manifolds of relative equilibria associated with the spatial symplectic structure the momentum of the connecting orbit or shape of the solitary wave considered as a heteroclinic orbit in a phase space representation is given a new characterization as a one form on the tangent space to the heteroclinic manifold and this one form is a restriction of the temporal symplectic structure for the linear stability analysis a new symplectic characterization of the evans function and its derivatives are obtained leading to an abstract geometric proof of instability for a large class of solitary wave states of equivariant hamiltonian evolutionary pdes the theory sheds new light on several well known models the gkdv equation a boussinesq system and a nonlinear wave equation the generalization to solitary waves associated with multidimensional heteroclinic manifolds and the implications for solitary waves or fronts which are biasymptotic to invariant manifolds such as periodic states are also discussed

numerical simulation of interaction between schrodinger field and klein gordon field by multisymplectic method abstract we propose a multisymplectic scheme to solve the coupled klein gordon schrodinger system the scheme preserves the multisymplectic geometry structure exactly by satisfying the discrete multisymplectic conservation law and can simulate the original waves well in a long time this scheme also has discrete quasi norm conservation law numerical experiments demonstrate the consistency between the theoretical analysis and the numerical results

global regularity criterion for the 3d navier stokes equations involving one entry of the velocity gradient tensor in this paper we provide a sufficient condition in terms of only one of the nine entries of the gradient tensor that is the jacobian matrix of the velocity vector field for the global regularity of strong solutions to the three dimensional navier stokes equations in the whole space as well as for the case of periodic boundary conditions

multi symplectic runge kutta collocation methods for hamiltonian wave equations a number of conservative pdes like various wave equations allow for a multi symplectic formulation which can be viewed as a generalization of the symplectic structure of hamiltonian odes we show that gauss legendre collocation in space and time leads to multi symplectic integrators i e to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for hamiltonian odes we also discuss the issue of conservation of energy and momentum since time discretization by a gauss legendre method is computational rather expensive we suggest several semi explicit multi symplectic methods based on gauss legendre collocation in space and explicit or linearly implicit symplectic discretizations in time

on the critical one component regularity for 3 d navier stokes system general case let us consider initial data v 0 for the homogeneous incompressible 3d navier stokes equation with vorticity belonging to l frac 32 cap l 2 we prove that if the solution associated with v 0 blows up at a finite time t star then for any p in 4 infty and any unit vector e of mathbb r 3 the lp norm in time with value in dot h frac 12 frac 2 p of v e mathbb r 3 blows up at t star

energy preserving schemes for nonlinear hamiltonian systems of wave equations application to the vibrating piano string the problem of the vibration of a string is well known in its linear form describing the transversal motion of a string nevertheless this description does not explain all the observations well enough nonlinear coupling between longitudinal and transversal modes seams to better model the piano string as does for instance the geometrically exact model gem this report introduces a general class of nonlinear systems nonlinear hamiltonian systems of wave equations in which fits the gem mathematical study of these systems is lead in a first part showing central properties energy preservation existence and unicity of a global smooth solution finite propagation velocity ldots space discretization is made in a classical way variational formulation and time discretization aims at numerical stability using an energy technique a definition of preserving schemes is introduced and we show that explicit schemes or partially implicit schemes which are preserving according to this definition cannot be built unless the model is linear a general energy preserving second order accurate fully implicit scheme is built for any continuous system that fits the nonlinear hamiltonian systems of wave equations class

conserved quantities of some hamiltonian wave equations after full discretization hamiltonian pdes have some invariant quantities which would be good to conserve with the numerical integration in this paper we concentrate on the nonlinear wave and schrodinger equations under hypotheses of regularity and periodicity we study how a symmetric space discretization makes that the space discretized system also has some invariants or nearly invariants which well approximate the continuous ones we conjecture some facts which would explain the good numerical approximation of them after time integration when using symplectic runge kutta methods or symmetric linear multistep methods for second order systems

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

accuracy of classical conservation laws for hamiltonian pdes under runge kutta discretizations we investigate conservative properties of runge kutta methods for hamiltonian partial differential equations it is shown that multi symplecitic runge kutta methods preserve precisely the norm square conservation law based on the study of accuracy of runge kutta methods applied to ordinary and partial differential equations we present some results on the numerical accuracy of conservation laws of energy and momentum for hamiltonian pdes under runge kutta discretizations

error propagation in numerical approximations near relative equilibria we study the propagation of errors in the numerical integration of perturbations of relative equilibrium solutions of hamiltonian differential equations with symmetries first it is shown that taking an initial perturbation of a relative equilibrium the corresponding solution is related in a first approximation to another relative equilibrium with the parameters perturbed from the original then this is used to prove that for stable relative equilibria error growth with respect to the perturbed solution is in general quadratic but only linear for schemes that preserve the invariant quantities of the problem in this sense the conclusion is similar to the one obtained when integrating unperturbed relative equilibria numerical experiments illustrate the results

on prodi serrin type conditions for the 3d navier stokes equations abstract in this paper we prove several new prodi serrin type regularity criteria with weak lebesgue integrability in both space and time for the 3d navier stokes equations in the whole space

a multisymplectic explicit scheme for the modified regularized long wave equation in this paper we derive a new 10 point multisymplectic scheme for the modified regularized long wave equation the new scheme is an explicit scheme in the sense that the third time level does not include nonlinear terms numerical results indicate that the new scheme not only provides satisfied numerical solutions but also preserves three invariants of motion very well

on the preservation of phase space structure under multisymplectic discretization in this paper we explore the local and global properties of multisymplectic discretizations based on finite differences and fourier spectral approximations multisymplectic ms schemes are developed for two benchmark nonlinear wave equations the sine gordon and nonlinear schrodinger equations we examine the implications of preserving the ms structure under discretization on the numerical scheme s ability to preserve phase space structure as measured by the nonlinear spectrum of the governing equation we find that the benefits of multisymplectic integrators include improved resolution of the local conservation laws dynamical invariants and complicated phase space structures

multi symplectic runge kutta nystrom methods for nonlinear schrodinger equations with variable coefficients in this paper we consider runge kutta nystrom rkn methods applied to nonlinear schrodinger equations with variable coefficients nlsevc concatenating symplectic nystrom methods in spatial direction and symplectic runge kutta methods in temporal direction for nlsevc leads to multi symplectic integrators i e to numerical methods which preserve the multi symplectic conservation law mscl we present the corresponding discrete version of mscl it is shown that the multi symplectic rkn methods preserve not only the global symplectic structure in time but also local and global discrete charge conservation laws under periodic boundary conditions we present a 4 order multi symplectic rkn method and use it in numerical simulation of quasi periodically solitary waves for nlsevc and we compare the multi symplectic rkn method with a non multi symplectic rkn method on the errors of numerical solutions the numerical errors of discrete energy discrete momentum and discrete charge the precise conservation of discrete charge under the multi symplectic rkn discretizations is attested numerically some numerical superiorities of the multi symplectic rkn methods are revealed

the life span of backward error analysis for numerical integrators backward error analysis is a useful tool for the study of numerical approximations to ordinary differential equations the numerical solution is formally interpreted as the exact solution of a perturbed differential equation given as a formal and usually divergent series in powers of the step size for a rigorous analysis this series has to be truncated in this article we study the influence of this truncation to the difference between the numerical solution and the exact solution of the perturbed differential equation results on the long time behaviour of numerical solutions are obtained in this way we present applications to the numerical phase portrait near hyperbolic equilibrium points to asymptotically stable periodic orbits and hopf bifurcation and to energy conservation and approximation of invariant tori in hamiltonian systems

conservation of energy momentum and actions in numerical discretizations of non linear wave equations for classes of symplectic and symmetric time stepping methods trigonometric integrators and the stormer verlet or leapfrog method applied to spectral semi discretizations of semilinear wave equations in a weakly non linear setting it is shown that energy momentum and all harmonic actions are approximately preserved over long times for the case of interest where the cfl number is not a small parameter such results are outside the reach of standard backward error analysis here they are instead obtained via a modulated fourier expansion in time

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

approximate momentum conservation for spatial semidiscretizations of semilinear wave equations we prove that a standard second order finite difference uniform space discretization of the semilinear wave equation with periodic boundary conditions analytic nonlinearity and analytic initial data conserves momentum up to an error which is exponentially small in the stepsize our estimates are valid for as long as the trajectories of the full semilinear wave equation remain real analytic the method of proof is that of backward error analysis whereby we construct a modified equation which is itself lagrangian and translation invariant and therefore also conserves momentum this modified equation interpolates the semidiscrete system for all time and we prove that it remains exponentially close to the trigonometric interpolation of the semidiscrete system these properties directly imply approximate momentum conservation for the semidiscrete system we also consider discretizations that are not variational as well as discretizations on non uniform grids through numerical example as well as arguments from geometric mechanics and perturbation theory we show that such methods generically do not approximately preserve momentum

on nonparaxial nonlinear schrodinger type equations abstract in this paper the one dimensional nonparaxial nonlinear schrodinger equation is considered this was proposed as an alternative to the classical nonlinear schrodinger equation in those situations where the assumption of paraxiality may fail the paper contributes to the mathematical properties of the equation in a two fold way first some theoretical results on linear well posedness hamiltonian and multi symplectic formulations are derived then we propose to take into account these properties in order to deal with the numerical approximation in this sense different numerical procedures that preserve the hamiltonian and multi symplectic structures are discussed and illustrated with numerical experiments

bilinear estimates in bmo and the navier stokes equations we prove that the bmo norm of the velocity and the vorticity controls the blow up phenomena of smooth solutions to the navier stokes equations our result is applied to the criterion on uniqueness and regularity of weak solutions in the marginal class

the exponential accuracy of fourier and chebyshev differencing methods it is shown that when differencing analytic functions using the pseudospectral fourier or chebyshev methods the error committed decays to zero at an exponential rate

numerical analysis of a multi symplectic scheme for a strongly coupled schrodinger system abstract in this paper we analyze a multi symplectic scheme for a strongly coupled nonlinear schrodinger equations we derive a general box scheme which is equivalent to the multi symplectic scheme by reduction method based on the general box scheme we prove that the multi symplectic scheme preserves not only the multi symplectic structure of the equation but also conservation law of mass in general the multi symplectic schemes are not conservative to energy in the nonlinear case so it is difficult to obtain the estimates of numerical solutions in norm hence proofs of convergence and stability are difficult for multi symplectic schemes of nonlinear equations a deduction argument and the energy analysis method are used to prove that the numerical solution is stable for initial values and second order convergent to the exact solutions in 2 norm a fixed point theorem is introduced and used to prove the unique solvability of the numerical solutions

numerical methods for hamiltonian pdes the paper provides an introduction and survey of conservative discretization methods for hamiltonian partial differential equations the emphasis is on variational symplectic and multi symplectic methods the derivation of methods as well as some of their fundamental geometric properties are discussed basic principles are illustrated by means of examples from wave and fluid dynamics

a technique to construct symmetric variable stepsize linear multistep methods for second order systems some previous works show that symmetric fixed and variable stepsize linear multistep methods for second order systems which do not have any parasitic root in their first characteristic polynomial give rise to a slow error growth with time when integrating reversible systems in this paper we give a technique to construct variable stepsize symmetric methods from their fixed stepsize counterparts in such a way that the former have the same order as the latter the order and symmetry of the integrators obtained is proved independently of the order of the underlying fixed stepsize integrators as this technique looks for efficiency we concentrate on explicit linear multistep methods which just make one function evaluation per step and we offer some numerical comparisons with other one step adaptive methods which also show a good long term behaviour

a geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities action symplecticity signature and complex instability are fundamental concepts in hamiltonian dynamics which can be characterized in terms of the symplectic structure in this paper hamiltonian pdes on unbounded domains are characterized in terms of a multisymplectic structure where a distinct differential two form is assigned to each space direction and time this leads to a new geometric formulation of the conservation of wave action for linear and nonlinear hamiltonian pdes and via stokes s theorem a conservation law for symplecticity each symplectic structure is used to define a signature invariant on the eigenspace of a normal mode the first invariant in this family is classical krein signature or energy sign when the energy is time independent and the other spatial signatures are energy flux signs leading to a classification of instabilities that includes information about directional spatial spreading of an instability the theory is applied to several examples the boussinesq equation the water wave equations linearized about an arbitrary stokes s wave rotating shallow water flow and flow past a compliant surface some implications for non conservative systems are also discussed

backward error analysis for numerical integrators backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods this is true in particular for the integration of hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time such results are typically based on two aspects of backward error analysis i it can be shown that the modified vector fields have some qualitative properties which they share with the given problem and ii an estimate is given for the difference between the best interpolating vector field and the numerical method these aspects have been investigated recently for example by benettin and giorgilli in j statist phys 74 1994 pp 1117 1143 by hairer in ann numer math 1 1994 pp 107 132 and by hairer and lubich in numer math 76 1997 pp 441 462 in this paper we aim at providing a unifying framework and a simplification of the existing results and corresponding proofs our approach to backward error analysis is based on a simple recursive definition of the modified vector fields that does not require explicit taylor series expansion of the numerical method and the corresponding flow maps as in the above cited works as an application we discuss the long time integration of chaotic hamiltonian systems and the approximation of time averages along numerically computed trajectories

invariant conserving finite difference algorithms for the nonlinear klein gordon equation a formalism for systematicaiiy deriving second order accurate finite difference algorithms which conserve certain invariant quantities in the original nonlinear pdes is presented three algorithms are derived for the nonlinear klein gordon equation nlkge based on the proposed formalism the local conservation laws of the nlkge form the basic starting point in our derivation which hinges essentially on the commutativity of certain finite difference operators such commutativity in the discrete approximations allows a preservation of the derivation properties of the continuous counterparts at the pde level with appropriate boundary conditions the proposed algorithms preserve in the discrete sense either the total system energy or the system s linear momentum several variants of the present algorithms and their relation to previously proposed algorithms are discussed an analysis of the accuracy and stability is conducted to compare the different variants of the proposed algorithms the preservation of energy of the present algorithms for the nlkge can also be viewed as providing a method of stabilization for conditionally stable algo thms for the linear wave equation the computer impiementation of the proposed algorithms with the treatment of the boundary conditions is presented in detail numerical examples are given concerning soliton collisions in the sine gordon equation the double sine gordon equation and the cb phi four equation the numerical results demonstrate that the present algorithms can preserve accurately up to 10 decimal digits the total system energy for a very coarse grid reliable algorithms for josephson junction models which contain dissipation damping mechanisms and driving bias current are obtained as direct by products of the proposed invariant conserving algorithms for the nlkge even though presented mainly for the 1 d case the proposed algorithms are generalizable to the 2 d and 3 d cases and to the case of complex valued nlkge

symplectic structure preserving integrators for the two dimensional gross pitaevskii equation for bec a b s t r a c t symplectic integrators have been developed for solving the two dimensional gross pitaevskii equation the equation is transformed into a hamiltonian form with symplectic structure then symplectic integrators including the midpoint rule and a splitting symplectic scheme are developed for treating this equation it is shown that the proposed codes fulfill the discrete charge conservation law furthermore the global error of the numerical solution is theoretically estimated the theoretical analysis is supported by some numerical simulations

derivation of the discrete conservation laws for a family of finite difference schemes abstract we present conservative schemes in finite differences applied to nonlinear odes and nonlinear klein gordon schrodinger and dirac field equations a systematic approach is used and an explicit derivation of the corresponding conservation laws is given

symplectic wavelet collocation method for hamiltonian wave equations this paper introduces a novel symplectic wavelet collocation method for solving nonlinear hamiltonian wave equations based on the autocorrelation functions of daubechies compactly supported scaling functions collocation method is conducted for the spatial discretization which leads to a finite dimensional hamiltonian system then appropriate symplectic scheme is employed for the integration of the hamiltonian system under the hypothesis of periodicity the properties of the resulted space differentiation matrix are analyzed in detail conservation of energy and momentum is also investigated various numerical experiments show the effectiveness of the proposed method

on the representation of integers by quadratic forms let n 4 and let q 2 z x1 xn be a non singular quadratic form when q is indefinite we provide new upper bounds for the least non trivial integral solution to the equation q 0 and when q is positive definite we provide improved upper bounds for the greatest positive integer k for which the equation q k is insoluble in integers despite being soluble modulo every prime power

multisymplectic schemes for strongly coupled schrodinger system in this paper two semi explicit multisymplectic schemes are derived for the strongly coupled schrodinger system based on the two new multisymplectic schemes we obtain a multisymplectic composition scheme which improves the accuracy in time the best merits of the present schemes are all implemented easily some numerical simulations are done for investigating nonlinear coupling and linear coupling numerical results indicate that the new multisymplectic composition scheme is effective

energy conservation issues in the numerical solution of the semilinear wave equation in this paper we discuss energy conservation issues related to the numerical solution of the semilinear wave equation as is well known this problem can be cast as a hamiltonian system that may be autonomous or not depending on the prescribed boundary conditions we relate the conservation properties of the original problem to those of its semi discrete version obtained by the method of lines subsequently we show that the very same properties can be transferred to the solutions of the fully discretized problem obtained by using energy conserving methods in the hbvms hamiltonian boundary value methods class similar arguments hold true for different types of hamiltonian partial differential equations e g the nonlinear schrodinger equation

least zero of a cubic form an effective search bound is established for the least non trivial integer zero of an arbitrary cubic form c in mathbb z x 1 ldots x n provided that n 17

mimetic finite difference methods for hamiltonian wave equations in 2d abstract in this paper we consider the numerical solution of the hamiltonian wave equation in two spatial dimensions we construct a two step procedure in which we first discretize the space by the mimetic finite difference mfd method and then we employ a standard symplectic scheme to integrate the semi discrete hamiltonian system derived the main characteristic of the mfd methods when applied to stationary problems is to mimic important properties of the continuous system this approach yields a full numerical procedure suitable to integrate hamiltonian problems a complete theoretical analysis of the method and some numerical simulations are developed in the paper

multi symplectic preissmann scheme for a high order wave equation of kdv type the high order wave equation of kdv type which describes many important physical phenomena has been investigated widely in last several decades here the new multi symplectic formulations for the high order wave equation of kdv type are presented and the associated local conservation laws are shown the multi symplectic discretization of each formulation is exemplified by the multi symplectic preissmann scheme the numerical experiments are given and the results verify the efficiency of the multi symplectic preissmann scheme

general local energy preserving integrators for solving multi symplectic hamiltonian pdes in this paper we propose and investigate a general approach to constructing local energy preserving algorithms which can be of arbitrarily high order in time for solving hamiltonian pdes this approach is based on the temporal discretization using continuous runge kutta type methods and the spatial discretization using pseudospectral methods or gauss legendre collocation methods the local energy conservation law of our new schemes is analyzed in detail the effectiveness of the novel local energy preserving integrators is demonstrated by coupled nonlinear schrodinger equations and 2d nonlinear schrodinger equations with external fields our new schemes are compared with some classical multi symplectic and symplectic schemes in numerical experiments the numerical results show the remarkable long term behaviour of our new schemes

finite difference calculus invariant structure of a class of algorithms for the nonlinear klein gordon equation in a previous work the authors have presented a formalism for deriving systematically invariant symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities this formalism is herein cast in the context of exact finite difference calculus the algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus in the same sense as that of the corresponding differential equation being derivable from its associated energy function a conserved quantity a clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities which are the consistent counterpart of the invariant quantities in the continuous case results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism and that preserve energy or linear momentum are discussed in the context of finite difference calculus some

symplectic integration of hamiltonian wave equations the numerical integration of a wide class of hamiltonian partial differential equations by standard symplectic schemes is discussed with a consistent hamiltonian approach we discretize the hamiltonian and the poisson structure separately then form the the resulting ode s the stability accuracy and dispersion of different explicit splitting methods are analyzed and we give the circumstances under which the best results can be obtained in particular when the hamiltonian can be split into linear and nonlinear terms many different treatments and examples are compared

on the hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms we reconsider the problem of the hamiltonian interpolation of symplectic mappings following moser s scheme we prove that for any mapping e analytic and e close to the identity there exists an analytic autonomous hamiltonian system he such that its time one mapping he differs from e by a quantity exponentially small in 1 e this result is applied in particular to the problem of numerical integration of hamiltonian systems by symplectic algorithms it turns out that when using an analytic symplectic algorithm of orders to integrate a hamiltonian systemk one actually follows exactly namely within the computer roundoff error the trajectories of the interpolating hamiltonian he or equivalently of the rescaled hamiltonian ke e 1he which differs fromk but turns out to be e5 close to it special attention is devoted to numerical integration for scattering problems

note on prodi serrin ladyzhenskaya type regularity criteria for the navier stokes equations in this article we prove new regularity criteria of the prodi serrin ladyzhenskaya type for the cauchy problem of the three dimensional incompressible navier stokes equations our results improve the classical lr 0 t ls regularity criteria for both velocity and pressure by factors of certain negative powers of the scaling invariant norms u l 3 and u h 1 2

analysis of variable stepsize linear multistep methods with special emphasis on symmetric ones in this paper we deal with several issues concerning variable stepsize linear multistep methods first we prove their stability when their fixed stepsize counterparts are stable and under mild conditions on the stepsizes and the variable coefficients then we prove asymptotic expansions on the considered tolerance for the global error committed using them we study the growth of error with time when integrating periodic orbits we consider strongly and weakly stable linear multistep methods for the integration of first order differential systems as well as those designed to integrate special second order ones we place special emphasis on the latter which are also symmetric because of their suitability when integrating moderately eccentric orbits of reversible systems for these types of methods we give a characterization for symmetry of the coefficients which allows their construction and provide some numerical results for them

gshmc an efficient method for molecular simulation the hybrid monte carlo hmc method is a popular and rigorous method for sampling from a canonical ensemble the hmc method is based on classical molecular dynamics simulations combined with a metropolis acceptance criterion and a momentum resampling step while the hmc method completely resamples the momentum after each monte carlo step the generalized hybrid monte carlo ghmc method can be implemented with a partial momentum refreshment step this property seems desirable for keeping some of the dynamic information throughout the sampling process similar to stochastic langevin and brownian dynamics simulations it is however ultimate to the success of the ghmc method that the rejection rate in the molecular dynamics part is kept at a minimum otherwise an undesirable zitterbewegung in the monte carlo samples is observed in this paper we describe a method to achieve very low rejection rates by using a modified energy which is preserved to high order along molecular dynamics trajectories the modified energy is based on backward error results for symplectic time stepping methods the proposed generalized shadow hybrid monte carlo gshmc method is applicable to nvt as well as npt ensemble simulations

accuracy and conservation properties in numerical integration the case of the korteweg de vries equation when numerically integrating time dependent differential equations it is often recommended to employ methods that preserve some of the invariant quantities mass energy etc of the problem being considered this recommendation is usually justified on the grounds that conservation of invariant quantities may ensure that the numerical solution possesses some important qualitative features however there are cases where schemes that preserve invariants are also advantageous in that they possess favourable error propagation mechanisms that render them superior from a quantitative point of view in the present paper we consider the korteweg de vries equation as a case study we show rigorously that for soliton problems and at leading order the error of conservative schemes consists of a phase error that grows linearly with time plus a complementary term that is bounded in the h 1 norm uniformly in time for general nonconservative schemes the error involves a linearly growing amplitude error a quadratically growing phase error and a complementary term that grows linearly in the h 1 norm numerical experiments are presented

quadratic invariants and multi symplecticity of partitioned runge kutta methods for hamiltonian pdes in this paper we study the preservation of quadratic conservation laws of runge kutta methods and partitioned runge kutta methods for hamiltonian pdes and establish the relation between multi symplecticity of runge kutta method and its quadratic conservation laws for schrodinger equations and dirac equations it reveals that multi symplectic runge kutta methods applied to equations with appropriate boundary conditions can preserve the global norm conservation and the global charge conservation respectively

symmetric multistep methods over long times for computations of planetary motions with special linear multistep methods an excellent long time behaviour is reported in the literature without a theoretical explanation neither the total energy nor the angular momentum exhibit secular error terms in this paper we completely explain this behaviour by studying the modified equation of these methods and by analyzing the remarkably stable propagation of parasitic solution components

normal form and long time analysis of splitting schemes for the linear schrodinger equation with small potential we consider the linear schrodinger equation on a one dimensional torus and its time discretization by splitting methods assuming a non resonance condition on the stepsize and a small size of the potential we show that the numerical dynamics can be reduced over exponentially long time to a collection of two dimensional symplectic systems for asymptotically large modes for the numerical solution this implies the long time conservation of the energies associated with the double eigenvalues of the free schrodinger operator the method is close to standard techniques used in finite dimensional perturbation theory but extended here to infinite dimensional operators

on the stability of symplectic and energy momentum algorithms for non linear hamiltonian systems with symmetry abstract this paper presents a detailed comparison of two implicit time integration schemes for a simple non linear hamiltonian system with symmetry the motion of a particle in a central force field the goal is to establish analytical and numerical results pertaining to the stability properties of the implicit mid point rule the proto typical implicit symplectic method and a particular energy momentum conserving scheme and to compare the two schemes with respect to accuracy while all results presented herein are within the context of a simple model problem the problem was constructed so as to exhibit key features typical of more complex systems with symmetry such as those arising in non linear solid mechanics namely the presence of large and relatively slow overall motions together with high frequency internal motions

a symplectic procedure for two dimensional coupled elastic wave equations using radial basis functions interpolation abstract this study proposes a meshless symplectic algorithm for two dimensional coupled elastic waves which satisfy the equations of linear isotropic elasticity on a rectangular domain with appropriate initial and boundary conditions this method is based on a combination of radial basis functions rbfs collocation and geometric integrators the method involves two steps first the spatial discretization by using rbfs interpolation transforms the seismic waves into a finite dimensional hamiltonian system then evolves the semi discretized system by using symplectic integrators furthermore the conservations of energy and symplectic structure are investigated the truncation error and global error of the method are estimated numerical experiments show that the scheme is efficient easy to implement with the scattered knots and possesses a good behavior in the long time integration