time finite element methods a unified framework for numerical discretizations of odes abstract we present a unified framework for the numerical discretization of odes based on time finite element methods we relate time finite element methods to runge kutta methods with infinitely many stages by means of the corresponding numerical quadrature we establish the relation between time finite element methods and partitioned runge kutta methods we also provide order estimates and superconvergence of the corresponding numerical methods in use of the simplifying assumptions we apply time finite methods to hamiltonian systems and investigate the conservation of energy and symplectic structure for the resulting numerical discretizations for hamiltonian systems we also construct new classes of symplectic integrators by combining different time finite element methods and verify the results by performing some numerical experiments

multisymplectic geometry and multisymplectic preissman scheme for the kp equation the multisymplectic structure of the kp equation is obtained directly from the variational principal using the covariant de donder weyl hamilton function theories we reformulate the kp equation to the multisymplectic form which was proposed by bridges from the multisymplectic equation we can derive a multisymplectic numerical scheme of the kp equation which can be simplified to the multisymplectic 45 points scheme

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

on the vanishing viscosity limit for the 3d navier stokes equations with a slip boundary condition where and below and denote the div and curl operators respectively n is the outward normal and is the unit tangential vector of the investigation of vanishing viscosity limit of solutions of the navier stokes equations both in the two and three spacial dimensional cases is a classical issue there are two related questions arising from here one is how to describe the inviscid limiting behavior of the navier stokes equation and the other is that does the euler equation can be approximated by the navier stokes equations in the case that the solution to the this research is supported in part by zheng ge ru foundation and hong kong rgc earmarked research grants cuhk 4028 04p cuhk 4040 02p and cuhk 4279 00p

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

an algebraic approach to invariant preserving integators the case of quadratic and hamiltonian invariants in this article conditions for the preservation of quadratic and hamiltonian invariants by numerical methods which can be written as b series are derived in a purely algebraical way the existence of a modified invariant is also investigated and turns out to be equivalent up to a conjugation to the preservation of the exact invariant a striking corollary is that a symplectic method is formally conjugate to a method that preserves the hamitonian exactly another surprising consequence is that the underlying one step method of a symmetric multistep scheme is formally conjugate to a symplectic p series when applied to newton s equations of motion

a roadmap for discretely energy stable schemes for dissipative systems based on a generalized auxiliary variable with guaranteed positivity abstract we present a framework for devising discretely energy stable schemes for general dissipative systems based on a generalized auxiliary variable the auxiliary variable a scalar number can be defined in terms of the energy functional by a general class of functions not limited to the square root function adopted in previous approaches the current method has another remarkable property the computed values for the generalized auxiliary variable are guaranteed to be positive on the discrete level regardless of the time step sizes or the external forces this property of guaranteed positivity is not available in previous approaches a unified procedure for treating the dissipative governing equations and the generalized auxiliary variable on the discrete level has been presented the discrete energy stability of the proposed numerical scheme and the positivity of the computed auxiliary variable have been proved for general dissipative systems the current method termed gpav generalized positive auxiliary variable requires only the solution of linear algebraic equations within a time step with appropriate choice of the operator in the algorithm the resultant linear algebraic systems upon discretization involve only constant and time independent coefficient matrices which only need to be computed once and can be pre computed several specific dissipative systems are studied in relative detail using the gpav framework ample numerical experiments are presented to demonstrate the performance of the method and the robustness of the scheme at large time step sizes

reduced solutions of douglas equations and angles between subspaces in this paper we study some particular solutions of douglas type equations by means of generalized inverses and angles we apply this result to characterize positive solutions and some special projections which are symmetrizable for a semidefinited positive bounded linear operator

new extremal characterizations of generalized inverses of linear operators abstract in 1956 r penrose studied best approximate solutions of the matrix equation ax b he proved that a b where a is the moore penrose inverse is the unique matrix of minimal frobenius norm among all matrices which minimize the frobenius norm of ax b in particular a is the unique best approximate solution of ax i the vector version of penrose s result that is the fact that the vector a b is the best approximate solution in the euclidean norm of the vector equation ax b has long been generalized to infinite dimensional hilbert spaces in this paper an infinite dimensional version of penrose s full result is given we show that a straightforward generalization is not possible and provide new extremal characterizations in terms of the hermitian order of a and of the classes of generalized inverses associated with minimal norm solutions of consistent operator equations or with least squares solutions for a certain class of operators we can phrase our characterizations in terms of a whole class of norms including the hilbert schmidt and the trace norms thus providing new extremal characterizations even in the matrix case we treat both operators with closed range and with not necessarily closed range finally we characterize a as the unique inner inverse of minimal hilbert schmidt norm if a 2

galerkin finite element solution for benjamin bona mahony burgers equation with cubic b splines abstract in this article we study solitary wave solutions of the nonlinear benjamin bona mahony burgers bbm burgers equation based on a lumped galerkin technique using cubic b spline finite elements for the spatial approximation the existence and uniqueness of solutions of the galerkin version of the solutions have been established an accuracy analysis of the galerkin finite element scheme for the spatial approximation has been well studied the proposed scheme is carried out for four test problems including dispersion of single solitary wave interaction of two three solitary waves and development of an undular bore then we propose a full discrete scheme for the resulting ivp von neumann theory is used to establish stability analysis of the full discrete numerical algorithm to display applicability and durableness of the new scheme error norms l 2 l and three invariants i 1 i 2 and i 3 are computed and the acquired results are demonstrated both numerically and graphically the obtained results specify that our new scheme ensures an apparent and an operative mathematical instrument for solving nonlinear evolution equation

construction of symplectic partitioned runge kutta methods with continuous stage hamiltonian systems as one of the most important class of dynamical systems are associated with a well known geometric structure called symplecticity symplectic numerical algorithms which preserve such a structure are therefore of interest in this article we study the construction of symplectic partitioned runge kutta methods with continuous stage this construction of symplectic methods mainly relies upon the expansion of orthogonal polynomials and the simplifying assumptions for partitioned runge kutta type methods by using suitable quadrature formulae it also provides a new and simple way to construct symplectic partitioned runge kutta methods in classical sense

novel implicit explicit local conservative schemes for the regularized long wave equation and convergence analysis abstract two implicit and two explicit schemes preserving the local momentum conservation laws exactly on any time space region are proposed for the regularized long wave equation with appropriate boundary conditions the schemes will be energy and mass preserving globally combining with the momentum conservation laws we obtain the priori estimates of the numerical solution and the error estimates in l norm for the proposed implicit schemes numerical experiments show the excellent performance of the proposed schemes and the numerical results coincide with the theoretical ones

two classes of linearly implicit local energy preserving approach for general multi symplectic hamiltonian pdes abstract two classes of efficient and robust schemes are proposed for the general multi symplectic hamiltonian systems using the invariant energy quadratization ieq approach the schemes are linear second order accurate local energy preserving and preserve the global energy they are not restricted to specific forms of the nonlinear part of the state function and only require solving linear equations at each time step we applied the new schemes to various multi symplectic hamiltonian pdes to demonstrate their effectiveness computational efficiency and accuracy

special issue on geometric numerical integration of differential equations this is a call for contributions to a special issue of journal of physics a mathematical and general entitled geometric numerical integration of differential equations this issue should be a repository for high quality original work we are interested in having the topic interpreted broadly that is to include contributions dealing with symplectic or multisymplectic integration volume preserving integration symmetry preserving integration integrators that preserve first integrals lyapunov functions or dissipation exponential integrators integrators for highly oscillatory systems lie group integrators etc papers on geometric integration of both odes and pdes will be considered as well as application to molecular scale integration celestial mechanics particle accelerators fluid flows population models epidemiological models and or any other areas of science we believe that this issue is timely and hope that it will stimulate further development of this new and exciting field the editorial board has invited g r w quispel and r i mclachlan to serve as guest editors for the special issue their criteria for acceptance of contributions are the following the subject of the paper should relate to geometric numerical integration in the sense described above contributions will be refereed and processed according to the usual procedure of the journal papers should be original reviews of a work published elsewhere will not be accepted the guidelines for the preparation of contributions are as follows the deadline for submission of contributions is 1 september 2005 this deadline will allow the special issue to appear in late 2005 or early 2006 there is a strict page limit of 16 printed pages approximately 9600 words per contribution for papers exceeding this limit the guest editors reserve the right to request a reduction in length further advice on publishing your work in journal of physics a mathematical and general may be found at www iop org journals jphysa contributions to the special issue should if possible be submitted electronically by web upload at www iop org journals jphysa or by e mail to jphysa iop org quoting jphysa special issue geometric integration submissions should ideally be in standard latex form we are however able to accept most formats including microsoft word please see the web site for further information on electronic submissions authors unable to submit electronically may send hard copy contributions to publishing administrators journal of physics a institute of physics publishing dirac house temple back bristol bs1 6be uk enclosing the electronic code on floppy disk if available and quoting jphysa special issue geometric integration all contributions should be accompanied by a read me file or covering letter giving the postal and e mail addresses for correspondence the publishing office should be notified of any subsequent change of address this special issue will be published in the paper and online version of the journal the corresponding author of each contribution will receive a complimentary copy of the issue

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

on the roughness induced effective boundary conditions for an incompressible viscous flow abstract we consider the laminar viscous channel flow with the lateral surface of the channel containing surface irregularities it is supposed that a uniform pressure gradient is maintained in the longitudinal direction of the channel after studying the corresponding boundary layers we obtain rigorously the navier friction condition it is valid when the size and amplitude of the imperfections tend to zero furthermore the coefficient in the law is determined through an auxiliary boundary layer type problem and the tangential drag force and the effective mass flow are determined up to order o e3 2 the value of the effective coefficient is shown to be independent with respect to the position of the mean surface in the range of o e

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

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

factorizations of linear relations abstract given two linear relations a and b we characterize the existence of a linear relation operator c such that a b c respectively a c b these factorizations extend and improve well known results by douglas and sebestyen

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

weighted least squares solutions of the equation axb c 0 abstract let h be a hilbert space l h the algebra of bounded linear operators on h and w l h a positive operator such that w 1 2 is in the p schatten class for some 1 p given a b l h with closed range and c l h we study the following weighted approximation problem analyze the existence of 0 1 m i n x l h a x b c p w where x p w w 1 2 x p we also study the related operator approximation problem analyze the existence of 0 2 m i n x l h a x b c w a x b c where the order is the one induced in l h by the cone of positive operators in this paper we prove that the existence of the minimum of 0 2 is equivalent to the existence of a solution of the normal equation a w a x b c 0 we also give sufficient conditions for the existence of the minimum of 0 1 and we characterize the operators where the minimum is attained

solitary waves of the regularized long wave equation abstract a finite element solution of the regularised long wave equation based on galerkin s method using cubic splines as element shape functions is set up a linear stability analysis shows the scheme to be unconditionally stable test problems including the migration and interaction of solitary waves are used to validate the method which is found to be accurate and efficient the three invariants of the motion are evaluated to determine the conservation properties of the algorithm the temporal evolution of a maxwellian initial pulse is then studied

a new explicit multisymplectic scheme for the regularized long wave equation in this paper we derive a new ten point multisymplectic scheme for the regularized long wave equation from its bridges multisymplectic form the new scheme is an explicit scheme in the sense that it does not need iteration we discuss some properties of the new scheme the performance and the efficiency of the new scheme are illustrated by solving several test examples the obtained results are presented and compared with previous methods numerical results indicate that the multisymplectic scheme cannot only obtain satisfied solutions for the regularized long wave equation but also keep three invariants of motion which are evaluated to determine the conservation properties of the algorithm very well

finite element discretization of the navier stokes equations with a free capillary surface the instationary navier stokes equations with a free capillary boundary are considered in 2 and 3 space dimensions a stable finite element discretization is presented the key idea is the treatment of the curvature terms by a variational formulation in the context of a discontinuous in time space time element discretization stability in weak energy norms can be proved numerical examples in 2 and 3 space dimensions are given

improving the accuracy of the avf method the average vector field avf method is a b series scheme of the second order as a discrete gradient method it preserves exactly the energy integral for any canonical hamiltonian system we present and discuss two locally exact and energy preserving modifications of the avf method avf lex of the third order and avf slex of the fourth order applications to spherically symmetric potentials are given including a compact explicit expression for the avf scheme for the coulomb kepler problem

a lyapunov type theorem for dissipative numerical integrators with adaptive time stepping the asymptotic behavior of continuous dissipative systems and dissipative numerical integrators with fixed time stepping can be fully investigated by a lyapunov type theorem on continuous and discrete dynamical systems respectively however once adaptive time stepping is involved such theories cease to work and usually the dynamics should be investigated from the past instead of the standard forward way such as in terms of pullback attractors in this paper we present a different approach we stick to a forward definition of limit sets and show that still we can establish a lyapunov type theorem which reveals the precise asymptotic behavior of adaptive time stepping integrators in the presence of a discrete lyapunov functional

application of cubic b splines for numerical solution of the rlw equation cubic b spline functions have been used to develop a collocation method to solve the regularized long wave rlw equation which is used to model solitary waves undular bore development and wave generation performance of the scheme is tested by computing a solitary wave solution of the rlw equation and made comparison with analytical results undular bore development and wave generation is shown to be in good agreement with available results

numerical integrators that preserve symmetries and reversing symmetries we consider properties of flows the relationships between them and whether numerical integrators can be made to preserve these properties this is done in the context of automorphisms and antiautomorphisms of a certain group generated by maps associated to vector fields this new framework unifies several known constructions we also use the concept of covariance of a numerical method with respect to a group of coordinate transformations the main application is to explore the relationship between spatial symmetries reversing symmetries and time symmetry of flows and numerical integrators

an introduction to lie group integrators basics new developments and applications we give a short and elementary introduction to lie group methods a selection of applications of lie group integrators are discussed finally a family of symplectic integrators on cotangent bundles of lie groups is presented and the notion of discrete gradient methods is generalised to lie groups

solving the generalized regularized long wave equation on the basis of a reproducing kernel space on the basis of a reproducing kernel space an iterative algorithm for solving the generalized regularized long wave equation is presented the analytical solution in the reproducing kernel space is shown in a series form and the approximate solution u n is constructed by truncating the series to n terms the convergence of u n to the analytical solution is also proved results obtained by the proposed method imply that it can be considered as a simple and accurate method for solving such evolution equations

some linearly and non linearly implicit schemes for the numerical solutions of the regularized long wave equation in this paper we present some linearly implicit and non linearly implicit schemes for investigating the numerical solutions of the regularized long wave equation numerical experiments indicate that all the present schemes can give satisfactory numerical solutions meanwhile numerical results also confirm the excellent conservations of three invariants in long time computations

petrov galerkin methods for nonlinear dispersive waves abstract petrov galerkin methods based on piecewise linear interpolants for the korteweg de vries and related equations are studied both accuracy and stability are analysed for the linearised case it is shown that in the nonlinear case the order of accuracy of the standard galerkin procedure is reduced and an alternative technique is therefore proposed which retains the fourth order accuracy in space this is found to perform well when compared with finite difference or other finite element schemes

application of the variational iteration method to the regularized long wave equation this paper applies the variational iteration method to the regularized long wave equation in this method a generalized lagrange multiplier is introduced to construct a correction functional for the discussed problem the multiplier can be identified optimally via variational theory three invariants of the motion are evaluated to determine the conservation properties of the system

a computational method for regularized long wave equation abstract quintic spline technique and splitting method have been used to develop a new finite difference method to solve regularized long wave rlw equation the convergence and the stability of the proposed method are discussed then it is used to model solitary wave motion and undular bore development by solving two test examples

efficient energy preserving integrators for oscillatory hamiltonian systems in this paper we focus our attention on deriving and analyzing an efficient energy preserving formula for the system of nonlinear oscillatory or highly oscillatory second order differential equations q t mq t f q t where m is a symmetric positive semi definite matrix with m 1 and f q q u q is the negative gradient of a real valued function u q this system is a hamiltonian system with the hamiltonian h p q 1 2 p t p 1 2 q t mq u q the energy preserving formula exactly preserves the hamiltonian we analyze in detail the properties of the energy preserving formula and propose new efficient energy preserving integrators in the sense of numerical implementation the convergence analysis of the fixed point iteration is presented for the implicit integrators proposed in this paper it is shown that the convergence of implicit average vector field methods is dependent on m whereas the convergence of the new energy preserving integrators is independent of m the fermi pasta ulam problem and the sine gordon equation are carried out numerically to show the competence and efficiency of the novel integrators in comparison with the well known average vector field methods in the scientific literature

operator theoretic and computational approaches to ill posed problems with applications to antenna theory a general framework for regularization and approximation methods for ill posed problems is developed three levels in the resolution processes are distinguished and emphasized philosophy of resolution regularization approximation schema and regularization algorithms dilemmas and methodologies of resolution of ill posed problems and their numerical implementations are examined in this framework with particular reference to the problem of finding numerically minimum weighted norm least squares solutions of first kind integral equations and more generally of linear operator equations with nonclosed range a common problem in all these methods is delineated each method reduces the problem of resolution to a nonstandard minimization problem involving an unknown critical parameter whose optimal value is crucial to the numerical realization and amenability of the method the nonstandardness results from the fact that one does not have explicitly or a priori the function to be minimized it has to built up using additional information convergence rate estimates and robustness conditions etc several results are developed that complement recent advances in numerical analysis and regularization of inverse and ill posed identification and pattern synthesis problems an emphasis is placed on the role of constraints function space methods the role of generalized inverses and reproducing kernels in the regularization and stable computational resolution of these problems the results will be applied specifically to problems of antenna synthesis and identification however the thrust of the paper is devoted to the interdisciplinary character of operator theoretic and numerical methods for ill posed problems

long time accuracy of lie trotter splitting methods for langevin dynamics a new characterization of sufficient conditions for the lie trotter splitting to capture the numerical invariant measure of nonlinear ergodic langevin dynamics up to an arbitrary order is discussed our characterization relies on backward error analysis and needs weaker assumptions than assumed so far in the literature in particular neither high weak order of the splitting scheme nor symplecticity are necessary to achieve high order approximation of the invariant measure of the langevin dynamics numerical experiments confirm our theoretical findings

couette flows over a rough boundary and drag reduction we consider the couette flow between two plates the lower plate is fixed and has periodically placed riblets of the characteristic size on it in the limit 0 we find the effective couette navier flow as an o 2 approximation for the effective mass flow and an o 2 l 1 approximation for the velocity in the effective solution the effect of roughness enters through the navier slip condition with the matrix coefficient in front of the effective shear stress calculated using a boundary layer problem furthermore an o 2 approximation for the tangential drag force is found in all estimates explicit dependence on the kinematic viscosity the velocity of the upper plate and the distance between the plates l 3 is kept also the uniqueness of the solution is expressed through a non linear algebraic condition linking and l 3 then the result is applied to the viscous sub layers around immersed bodies strictly containing the surface riblets it is found that for the riblets of the characteristic size being of the order smaller or equal to the approximation obtained for the tangential drag could be applied we compare and for realistic data and our results lead to the conclusion that the riblets reduce significantly tangential drag which may explain their presence on the skin of nektons

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 meshfree method for the numerical solution of the rlw equation in this paper we present a meshfree technique for the numerical solution of the regularized long wave rlw equation this approach is based on a global collocation method using the radial basis functions rbfs different kinds of rbfs are used for this purpose accuracy of the new method is tested in terms of l 2 and l error norms in case of non availability of the exact solution performance of the new method is compared with existing methods stability analysis of the method is established propagation of single and double solitary waves wave undulation and conservation properties of mass energy and momentum of the rlw equation are discussed

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

a discrete line integral method of order two for the lorentz force system in this paper we apply the boole discrete line integral to solve the lorentz force system which is written as a non canonical hamiltonian system the method is exactly energy conserving for polynomial hamiltonians of degree 4 in any other case the energy can also be conserved approximatively with comparison to well used boris method numerical experiments are presented to demonstrate the energy preserving property of the method

a computational method for the regularized long wave rlw equation a numerical solution of the regularized long wave rlw equation based on collocation method using cubic b spline finite elements is used to simulate the migration and interaction of solitary waves interaction of solitary waves with different amplitudes are shown the three invariants of the motion are evaluated to determine the conservation properties of the system the temporal evaluation of a maxwellian initial pulse is then studied and then we prove that the number of solitons which are generated from maxwellian initial condition are determined

an energy preserving and symmetric scheme for nonlinear hamiltonian wave equations in this chapter we derive and analyse an energy preserving and symmetric scheme for nonlinear hamiltonian wave equations which can exactly preserve the energy of the underlying hamiltonian wave equations to this end we first define and discuss the bounded operator argument functions on the underlying domain we then introduce an operator variation of constants formula based on which we present an energy preserving scheme for nonlinear hamiltonian wave equations the scheme preserves the energy of the original continuous hamiltonian system exactly in comparison with the existing work on this topic such as the well known average vector field avf formula for hamiltonian ordinary differential equations the energy preserving scheme avoids the semi discretisation of spatial derivatives and exactly preserves the hamiltonian of the original continuous hamiltonian wave equation this point is very significant in comparison with the avf formula since the avf formula can preserve only the energy of hamiltonian ordinary differential equations hence the main theme of this chapter is to establish a scheme which can exactly preserve the energy of the nonlinear hamiltonian wave equation the chapter is also accompanied by some examples

exponential integrators with quadratic energy preservation for linear poisson systems abstract motivated by the recent paper wang and wu 2018 12 where the authors proposed a class of functionally fitted energy preserving integrators for poisson systems in this paper we aim at developing exponential integrators with quadratic energy preservation for the important class of linear poisson systems by rewriting the linear poisson system in a new formulation we apply the symplectic exponential runge kutta methods and thus obtain a novel class of exponential integrators for the linear poisson system it is proved that the obtained exponential integrator can exactly preserve the quadratic energy in comparison with the functionally fitted energy preserving methods whose numerical performance depends on the fitted frequency the new exponential integrators are independent of the fitted frequency and convenient to use furthermore we investigate the convergence of the fixed point iteration used in the implementation of the proposed exponential integrator and show that the convergence region of the stepsize is independent of the norm of the skew symmetric matrix finally numerical results with the euler equations of a free rigid body soundly support the remarkable performance of the exponential integrator in both the global errors and the energy preservation

application of a lumped galerkin method to the regularized long wave equation in this paper a lumped galerkin method based on quadratic b spline finite elements is used to find numerical solutions of the one dimensional regularized long wave rlw equation with a variant of initial and boundary conditions the obtained numerical results show that the present method is a remarkably successful numerical technique for solving the equation results are compared with published numerical solutions a linear stability analysis of the scheme is also investigated

the discrete variational derivative method based on discrete differential forms as is well known for pdes that enjoy a conservation or dissipation property numerical schemes that inherit this property are often advantageous in that the schemes are fairly stable and give qualitatively better numerical solutions in practice lately furihata and matsuo have developed the so called discrete variational derivative method that automatically constructs energy preserving or dissipative finite difference schemes although this method was originally developed on uniform meshes the use of non uniform meshes is of importance for multi dimensional problems on the other hand the theories of discrete differential forms have received much attention recently these theories provide a discrete analogue of the vector calculus on general meshes in this paper we show that the discrete variational derivative method and the discrete differential forms by bochev and hyman can be combined applications to the cahn hilliard equation and the klein gordon equation on triangular meshes are provided as demonstrations we also show that the schemes for these equations are h1 stable under some assumptions in particular one for the nonlinear klein gordon equation is obtained by combination of the energy conservation property and the discrete poincare inequality which are the temporal and spacial structures that are preserved by the above methods

a conservative weighted finite difference scheme for regularized long wave equation in this paper a new finite difference method for initial boundary value problem of regularized long wave rlw equation is considered two energy conservative finite difference schemes are proposed and the prior estimate of the solution is obtained existence and uniqueness of numerical solution are derived it is proved that the finite difference schemes are convergent and stable numerical experiment results show that the theory is accurate and the schemes are efficient and reliable

parametric spline solution of the regularized long wave equation in this paper we present a numerical method based on parametric quintic splines for the regularized long wave rlw equation the truncation error is analyzed and the method shows that by choosing suitably parameters we can obtain various accuracy schemes stability analysis of the method is studied and the numerical results show that the method is unconditionally stable the efficiency of the method is examined by evaluating the error norms and conservation properties of mass energy and momentum the numerical simulations can validate and demonstrate the advantages of the method

symmetric integrators based on continuous stage runge kutta nystrom methods for reversible systems abstract in this paper we study symmetric integrators for solving second order ordinary differential equations on the basis of the notion of continuous stage runge kutta nystrom methods the construction of such methods heavily relies on the legendre expansion technique in conjunction with the symmetric conditions and simplifying assumptions for order conditions new families of symmetric integrators as illustrative examples are presented for comparing the numerical behaviors of the presented methods some numerical experiments are also reported

effective boundary conditions for laminar flows over periodic rough boundaries effective boundary conditions or wall laws are proposed for a laminar flow over a rough wall with periodic roughness elements these effective conditions are posed on a regularized boundary which allows the details of the wall to be avoided and dramatically reduces the computational cost the effective conditions stem from an asymptotic expansion of the solution which is presented here both first and second order conditions are discussed and tested numerically on bidimensional cases

explicit multistep method for the numerical solution of rlw equation abstract in this paper we present explicit multistep method for regularized long wave rlw equation the discretization in space is based on the galerkin method using linear space finite elements and for the time discretization the explicit multistep method is used propagation of solitary waves and interaction of two solitary waves are simulated using the proposed method to confirm the theoretical results

high order symplectic integrators based on continuous stage runge kutta nystrom methods abstract on the basis of the previous work by tang and zhang 37 in this paper we present a more effective way to construct high order symplectic integrators for solving second order hamiltonian equations instead of analyzing order conditions step by step as shown in the previous work the new technique of this paper is using legendre expansions to deal with the simplifying assumptions for order conditions with the new technique high order symplectic integrators can be conveniently devised by truncating an orthogonal series

exponential collocation methods for conservative or dissipative systems abstract in this paper we propose and analyse a novel class of exponential collocation methods for solving conservative or dissipative systems based on exponential integrators and collocation methods it is shown that these novel methods can be of arbitrarily high order and exactly or approximately preserve first integrals or lyapunov functions we also consider order estimates of the new methods furthermore we explore and discuss the application of our methods in important stiff gradient systems and it turns out that our methods are unconditionally energy diminishing and strongly damped even for very stiff gradient systems practical examples of the new methods are derived and the efficiency and superiority are confirmed and demonstrated by three numerical experiments including a nonlinear schrodinger equation as a byproduct of this paper arbitrary order trigonometric rkn collocation methods are also presented and analysed for second order highly oscillatory general systems the paper is accompanied by numerical results that demonstrate the great potential of this work

energy and quadratic invariants preserving integrators based upon gauss collocation formulae we introduce a new family of symplectic integrators for canonical hamiltonian systems each method in the family depends on a real parameter alpha when alpha 0 we obtain the classical gauss collocation formula of order 2s where s denotes the number of the internal stages for any given non null alpha the corresponding method remains symplectic and has order 2s 2 hence it may be interpreted as an o h 2s 2 symplectic perturbation of the gauss method under suitable assumptions we show that the parameter alpha may be properly tuned at each step of the integration procedure so as to guarantee energy conservation in the numerical solution as well as to maintain the original order 2s as the generating gauss formula

the lack of continuity and the role of infinite and infinitesimal in numerical methods for odes the case of symplecticity when numerically integrating canonical hamiltonian systems the long term conservation of some of its invariants for example the hamiltonian function itself assumes a central role the classical approach to this problem has led to the definition of symplectic methods among which we mention gauss legendre collocation formulae indeed in the continuous setting energy conservation is derived from symplecticity via an infinite number of infinitesimal contact transformations however this infinite process cannot be directly transferred to the discrete setting by following a different approach in this paper we describe a sequence of methods sharing the same essential spectrum and then the same essential properties which are energy preserving starting from a certain element of the sequence on i e after a finite number of steps

unified approach to hamiltonian systems poisson systems gradient systems and systems with lyapunov functions or first integrals systems with a first integral i e constant of motion or a lyapunov function can be written as linear gradient systems dot x l x nabla v x for an appropriate matrix function l with a generalization to several integrals or lyapunov functions the discrete time analogue delta x delta t l bar nabla v where bar nabla is a discrete gradient preserves v as an integral or lyapunov function respectively

decoupled local global energy preserving schemes for the n coupled nonlinear schrodinger equations abstract we develop two local energy preserving integrators and a global energy preserving integrator for the general multisymplectic hamiltonian system when applied to the 1d and multi dimensional n coupled nonlinear schrodinger equations the given schemes have the exact preservation of the local global conservation law and are decoupled in the components n n 1 2 n i e each of the components can be solved independently the decoupled feature is significant and helpful for overcoming the computational difficulty of the n coupled n 3 nonlinear schrodinger equations especially of the multi dimensional case the composition method is employed to improve the accuracy of the schemes in time and the discrete fast fourier transform is used to reduce the computational complexity several numerical experiments are carried out to exhibit the behaviors of the wave solutions numerical results confirm the theoretical results

weighted procrustes problems abstract let h be a hilbert space l h the algebra of bounded linear operators on h and w l h a positive operator such that w 1 2 is in the p schatten class for some 1 p given a l h with closed range and b l h we study the following weighted approximation problem analyze the existence of m i n x l h a x b p w where x p w w 1 2 x p in this paper we prove that the existence of this minimum is equivalent to a compatibility condition between r b and r a involving the weight w and we characterize the operators which minimize this problem as w inverses of a in r b

a linearly implicit energy preserving exponential integrator for the nonlinear klein gordon equation abstract in this paper we generalize the exponential energy preserving integrator proposed in the recent paper siam j sci comput 38 2016 a1876 a1895 for conservative systems which now becomes linearly implicit by further utilizing the idea of the scalar auxiliary variable approach comparing with the original exponential energy preserving integrator which usually leads to a nonlinear algebraic system our new method only involves a linear system with a constant coefficient matrix taking the nonlinear klein gordon equation and the nonlinear schrodinger equation for examples we derive the concrete energy preserving schemes and demonstrate their high efficiency through numerical experiments

two energy conserved splitting methods for three dimensional time domain maxwell s equations and the convergence analysis we devote the present paper to high accuracy energy preserving s avf 2 and s avf 4 schemes for the three dimensional time domain maxwell s equations based on the exponential operator splitting technique the fourier pseudospectral method and the averaged vector field method to obtain the present schemes the key is to propose the splitting methods for maxwell s equations in which all subsystems should hold the same hamiltonian the proposed schemes are energy preserving high order accurate and unconditionally stable while being implemented explicitly both schemes capture four energy invariants simultaneously rigorous error estimates of the schemes are established in the discrete l 2 norm the theoretical results show that the s avf 2 s avf 4 scheme converges with spectral accuracy in space and second order fourth order accuracy in time respectively numerical results support the theoretical analysis

numerical implementation of the multisymplectic preissman scheme and its equivalent schemes we analyze the multisymplectic preissman scheme for the kdv equation with the periodic boundary condition and show that the unconvergence of the widely used iterative methods to solve the resulting nonlinear algebra system of the preissman scheme is due to the introduced potential function a artificial numerical condition is added to the periodic boundary condition the added boundary condition makes the numerical implementation of the multisymplectic preissman scheme practical and is proved not to change the numerical solutions of the kdv equation based on our analysis we derive some new schemes which are not restricted by the artificial boundary condition and more efficient than the preissman scheme because of less computing cost and less computer storages by eliminating the auxiliary variables we also derive two schemes for the kdv equation one is a 12 point scheme and the other is an 8 point scheme as the byproducts we present two new explicit schemes which are not multisymplectic but still have remarkable numerical stable property numerical experiments on soliton collisions are also provided to confirm our conclusion and to show the benefits of the multisymplectic schemes with comparison of the spectral method and zabusky kruskal scheme

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

global well posedness for the lake equations abstract we prove global well posedness for the lake equations these equations arise to leading order in a low aspect ratio low froude number i e low wave speed and very small wave amplitude expansion of the three dimensional incompressible euler equations in a basin with a free upper surface and a spatially varying bottom topography well posedness means that there exists a solution that it is unique and that it depends continuously on the data i e the initial condition and the bottom topography our approach follows the works of yudovitch and bardos in constructing the solutions as the inviscid limit of solutions to a system with artificial viscosity which is the analog of the navier stokes with respect to the euler equations one of the main assumptions in the present work is a nondegenerate bottom topography

hopf algebras renormalization and noncommutative geometry we explore the relation between the hopf algebra associated to the renormalization of qft and the hopf algebra associated to the ncg computations of tranverse index theory for foliations

a general framework for deriving integral preserving numerical methods for pdes a general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented in particular linearly implicit methods preserving a time discretized version of the invariant are developed for systems of partial differential equations with polynomial nonlinearities the framework is rather general and allows for an arbitrary number of dependent and independent variables with derivatives of any order it is proved formally that second order convergence is obtained the procedure is applied to a test case and numerical experiments are provided

quantum enveloping algebras with von neumann regular cartan like generators and the pierce decomposition quantum bialgebras derivable from uq sl2 which contain idempotents and von neumann regular cartan like generators are introduced and investigated various types of antipodes invertible and von neumann regular on these bialgebras are constructed which leads to a hopf algebra structure and a von neumann hopf algebra structure respectively for them explicit forms of some particular r matrices also invertible and von neumann regular are presented and the latter respects the pierce decomposition

preserving algebraic invariants with runge kutta methods abstract we study runge kutta methods for the integration of ordinary differential equations and the retention of algebraic invariants as a general rule we derive two conditions for the retention of such invariants the first is a condition on the coefficients of the methods the second is a pair of partial differential equations that otherwise must be obeyed by the invariant this paper extends previous work on multistep methods in iserles technical report na1997 13 damtp university of cambridge 1997 the cases related to the retention of quadratic and cubic invariants perhaps of greatest relevance in applications are thoroughly discussed we conclude recommending a generalized class of runge kutta schemes namely lie group type runge kutta methods these are schemes for the solution of odes on lie groups but can be employed together with group actions to preserve a larger class of algebraic invariants without restrictions on the coefficients

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

local momentum preserving algorithms for the grlw equation abstract in this paper four local momentum preserving lmp algorithms for the grlw equation are studied the proposed algorithms which conserve the local momentum conservation in any local time space region are proved to be momentum and mass preserving globally with appropriate boundary conditions two of the algorithms are explicit and the other two are implicit we make nonlinear convergence and stability analysis for the implicit algorithms numerical experiments confirm the long time preservation of the proposed algorithms and verify the theoretical analysis

conservative scheme for a model of nonlinear dispersive waves and its solitary waves induced by boundary motion abstract a conservative difference scheme is given for a model of nonlinear dispersive waves convergence and stability of the scheme are proved by means of this scheme we explore numerically the relationship between the boundary data and the amplitudes and number of solitary waves it produces

geometric integrators for odes geometric integration is the numerical integration of a differential equation while preserving one or more of its geometric properties exactly i e to within round off error many of these geometric properties are of crucial importance in physical applications preservation of energy momentum angular momentum phase space volume symmetries time reversal symmetry symplectic structure and dissipation are examples in this paper we present a survey of geometric numerical integration methods for ordinary differential equations our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade to this end the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature

vanishing viscous limits for the 2d lake equations with navier boundary conditions abstract the vanishing viscosity limit is considered for the viscous lake equations with navier friction boundary conditions we prove that the inviscid limit satisfies the inviscid lake equations and the results include flows generated by l p initial vorticity with 1 p

a compact fourth order in space energy preserving method for riesz space fractional nonlinear wave equations in this work we investigate numerically a nonlinear hyperbolic partial differential equation with space fractional derivatives of the riesz type the model under consideration generalizes various nonlinear wave equations including the sine gordon and the nonlinear klein gordon models the system considered in this work is conservative when homogeneous dirichlet boundary conditions are imposed motivated by this fact we propose a finite difference method based on fractional centered differences that is capable of preserving the discrete energy of the system the method under consideration is a nonlinear implicit scheme which has various numerical properties among the most interesting numerical features we show that the methodology is consistent of second order in time and fourth order in space moreover we show that the technique is stable and convergent some numerical simulations show that the method is capable of preserving the energy of the discrete system this characteristic of the technique is in obvious agreement with the properties of its continuous counterpart