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

two energy conserving numerical schemes for the sine gordon equation two explicit conservative numerical schemes for the sine gordon equation are proposed their stability and convergence are proved numerical simulation of a sine gordon soliton shows that the new schemes are very accurate and fast

linear pdes and numerical methods that preserve a multisymplectic conservation law multisymplectic methods have recently been proposed as a generalization of symplectic ode methods to the case of hamiltonian pdes their excellent long time behavior for a variety of hamiltonian wave equations has been demonstrated in a number of numerical studies a theoretical investigation and justification of multisymplectic methods is still largely missing in this paper we study linear multisymplectic pdes and their discretization by means of numerical dispersion relations it is found that multisymplectic methods in the sense of bridges and reich phys lett a 284 2001 pp 184 193 and reich j comput phys 157 2000 pp 473 499 such as gauss legendre runge kutta methods possess a number of desirable properties such as nonexistence of spurious roots and conservation of the sign of the group velocity a certain cfl type restriction on delta t delta x might be required for methods higher than second order in time it is also demonstrated by means of the explicit midpoint method that multistep methods may exhibit spurious roots in the numerical dispersion relation for any value of delta t delta x despite being multisymplectic in the sense of discrete variational mechanics j e marsden g p patrick and s shkoller commun math phys 199 1999 pp 351 395

preserving energy resp dissipation in numerical pdes using the average vector field method we give a systematic method for discretizing hamiltonian partial differential equations pdes with constant symplectic structure while preserving their energy exactly the same method applied to pdes with constant dissipative structure also preserves the correct monotonic decrease of energy the method is illustrated by many examples in the hamiltonian case these include the sine gordon korteweg de vries nonlinear schrodinger linear time dependent schrodinger and maxwell equations in the dissipative case the examples are the allen cahn cahn hilliard ginzburg landau and heat equations

linear and hamiltonian conserving fourier pseudo spectral schemes for the camassa holm equation abstract in this paper we develop two linear conservative fourier pseudo spectral schemes for the camassa holm equation we first apply the fourier pseudo spectral method in space for the camassa holm equation to arrive at a spatial semi discretized system in which a corresponding discrete momentum conservation law is preserved then we employe the linear implicit crank nicolson scheme and the leap frog scheme for the semi discrete system respectively the two new fully discrete methods are proved to conserve the discrete momentum conservation law of the original system which implies the numerical solutions are bounded in the discrete l norm furthermore the proposed methods are unconditionally stable second order in time and high order in space and uniquely solvable numerical experiments are presented to show the convergence property as well as the efficiency and accuracy of the new schemes the proposed methods in this paper could be readily utilized to design linear momentum preserving numerical approximations for many other hamiltonian pdes

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

normal forms and versal deformations of linear hamiltonian systems abstract equivalence classes of time independent linear real hamiltonian systems can be identified up to canonical transformations with the orbits of the adjoint action of the real symplectic group on its lie algebra a new set of representatives also called normal forms for these orbits is given versal deformations of systems in normal form are constructed applications of versal deformations to the study of bifurcations of linear systems with small codimension are indicated

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

versal normal form for nonsemisimple singularities abstract the theory of versal normal form has been playing a role in normal form since the introduction of the concept by v i arnol d in 1 2 but there has been no systematic use of it that is in line with the semidirect character of the group of formal transformations on formal vector fields that is the linear part should be done completely first before one computes the nonlinear terms in this paper we address this issue by giving a complete description of a first order calculation in the case of the two and three dimensional irreducible nilpotent cases which is then followed up by an explicit almost symplectic calculation to find the transformation to versal normal form in a particular fluid dynamics problem and in the celestial mechanics l 4 problem

global well posedness for the generalized large scale semigeostrophic equations we prove existence and uniqueness of global classical solutions to the generalized large scale semigeostrophic equations with periodic boundary conditions this family of hamiltonian balance models for rapidly rotating shallow water includes the l1 model derived by r salmon in 1985 and its 2006 generalization by the second author the results are under the physical restriction that the initial potential vorticity is positive as strong as those available for the euler equations of ideal fluid flow in two dimensions moreover we identify a special case in which the velocity field is two derivatives smoother in sobolev space as compared to the general case our results are based on careful estimates which show that although the potential vorticity inversion is nonlinear bounds on the potential vorticity inversion operator remain linear in derivatives of the potential vorticity this permits the adaptation of an argument based on elliptic lp theory proposed by yudovich in 1963 for proving existence and uniqueness of weak solutions for the two dimensional euler equations to our particular nonlinear situation

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 unified approach to energy conservation and potential vorticity dynamics for arbitrarily structured c grids a numerical scheme applicable to arbitrarily structured c grids is presented for the nonlinear shallow water equations by discretizing the vector invariant form of the momentum equation the relationship between the nonlinear coriolis force and the potential vorticity flux can be used to guarantee that mass velocity and potential vorticity evolve in a consistent and compatible manner underpinning the consistency and compatibility of the discrete system is the construction of an auxiliary thickness equation that is staggered from the primary thickness equation and collocated with the vorticity field the numerical scheme also exhibits conservation of total energy to within time truncation error simulations of the standard shallow water test cases confirm the analysis and show convergence rates between 1st and 2nd order accuracy when discretizing the system with quasi uniform spherical voronoi diagrams the numerical method is applicable to a wide class of meshes including latitude longitude grids voronoi diagrams delaunay triangulations and conformally mapped cubed sphere meshes

some new structure preserving algorithms for general multi symplectic formulations of hamiltonian pdes many partial differential equations pdes can be written as a multi symplectic hamiltonian system which has three local conservation laws namely multi symplectic conservation law local energy conservation law and local momentum conservation law in this paper we give several systematic methods for discretizing general multi symplectic formulations of hamiltonian pdes including a local energy preserving algorithm a class of global energy preserving methods and a local momentum preserving algorithm the methods are illustrated by the nonlinear schrodinger equation and the korteweg de vries equation numerical experiments are presented to demonstrate the conservative properties of the proposed numerical methods

the stability of the lagrange triangular point and a theorem of arnold deprit and deprit bartholome 5 computed the normal form for the hamiltonian of the planar restricted three body problem at the lagrange equilateral triangle equilibrium point up to terms of order 4 this normalization was carried out for all mass ratios p smaller than routh s critical mass ratio p1 i 1 j 9 except for two values p2 and pi at p2 and u the ratio of the linearized frequencies is 1 2 and 1 3 respectively and so low order resonance terms appear this normalization was carried out in order to apply the kam theory to prove the stability of this equilibrium point from the coefficients of the normal form they computed a quantity d defined below which when non zero establishes the stability of the equilibrium point they found that d 0 for 0 cp p u p2 p and p the value u does not correspond to resonance but is simply a point where the quantity d4 changes sign in this paper we shall establish the full stability of the lagrange equilibrium point in the planar restricted three body problem even in the case when u p we do this by computing by machine the normal form up to terms of order 6 and then applying a theorem of arnold 3 which establishes the stability of the equilibrium even in degenerate cases russman lo announced a theorem which implies that the equilibrium is isoenergetic stable when p p by isoenergetic stable he means that the system is stabie only on the energy surface defined by the energy at the equilibrium itself unfortunately there is not even an indication of the proof arnold announced his theorem in a doklady note with only a sketch of a proof the sketch indicates that his proof requires completely redoing all

versal normal form at the lagrange equilibrium l4 a new normal form called versal for the linearized hamiltonian vector field of the planar restricted three body problem at the lagrange equilibrium point l4 depending smoothly on the mass ratio for all values close to the critical routh s ratio is described then a canonical transformation also depending smoothly on the mass ratio which brings the linear hamiltonian vector field into this versal normal form is explicitly calculated

versal normal form of the hamiltonian function of the restricted problem of three bodies near l 4 abstract when one studies matrices depending on parameters the transformation into jordan canonical form can become singular when multiple eigenvalues occur the versal normal form for a matrix was introduced by arnol d 1971 in order to overcome this difficulty unfortunately no method was given on how to construct this transformation for a special class of matrices we describe a simple procedure to accomplish this task we use the one to one resonance at the lagrangian triangular equilibrium point in the restricted problem of three bodies as an example to demonstrate our method we then extend the idea of versal normal form to higher order terms in the hamiltonian function after performing this normalization we are able to analyze the behavior of the periodic orbits near the equilibrium point

a note on multisymplectic fourier pseudospectral discretization for the nonlinear schrodinger equation based on the multisymplectic fourier pseudospectral scheme for the nonlinear schrodinger nls equation we investigate some discrete properties corresponding to local conservation laws of the original equation the discrete normal conservation law is proved and the error estimation of local and global energy conservation laws are also obtained numerical experiments for cubic nls equation are provided to demonstrate the consistency between the theoretical analysis and the numerical results

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

transformation to versal deformations of matrices abstract in the paper versal deformations of matrices are considered the versal deformation is a matrix family inducing an arbitrary multi parameter deformation of a given matrix by an appropriate smooth change of parameters and basis given a deformation of a matrix it is suggested to find transformation functions the change of parameters and the change of basis dependent on parameters in the form of taylor series the general method of construction of recurrent procedures for calculation of coefficients in the taylor expansions is developed and used for the cases of real and complex matrices elements of classical lie and jordan algebras and infinitesimally reversible matrices several examples are given and studied in detail applications of the suggested approach to problems of stability singularity and perturbation theories are discussed

a new class of energy preserving numerical integration methods the first ever energy preserving b series numerical integration method for ordinary differential equations is presented and applied to several hamiltonian systems related novel lie algebraic results are also discussed

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