This paper develops a formalism for the design of conserving time-integration schemes for Hamiltonian systems with symmetry. The main result is that, through the introduction of a discrete directional derivative, implicit second-order conserving schemes can be constructed for general systems which preserve the Hamiltonian along with a certain class of other first integrals arising from affine symmetries. Discrete Hamiltonian systems are introduced as formal abstractions of conserving schemes and are analyzed within the context of discrete dynamical systems; in particular, various symmetry and stability properties are investigated.

Time Integration and Discrete Hamiltonian Systems

We study the application of Runge-Kutta schemes to Hamiltonian systems of ordinary differential equations We investigate which schemes possess the canonical property of the Hamiltonian flow We also consider the issue of exact conservation in the time-discretization of the continuous invariants of motion

Runge-Kutta schemes for Hamiltonian systems

Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations

Dispersion-managed solitons in fibre systems and lasers

We approximate the solutions of an initial- and boundary-value problem for nonlinear Schrodinger equations (with emphasis on the `cubic' nonlinearity) by two fully discrete finite element schemes based on the standard Galerkin method in space and two implicit. Crank-Nicolson-type second-order accurate temporal discretizations. For both schemes we study the existence and uniqueness of their solutions and proveL 2 error bounds of optimal order of accuracy. For one of the schemes we also analyze one step of Newton's method for solving the nonlinear systems that arise at every time step. We then implement this scheme using an iterative modification of Newton's method that, at each time stept n , requires solving a number of sparse complex linear systems with a matrix that does not change withn. The effect of this `inner' iteration is studied theoretically and numerically.

On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation

Optimal L2 rates of convergence are established for several fully-discrete schemes for the numerical solution of the nonlinear Schroedinger equation. Both finite differences and finite elements are considered for the discretization in space, while the integration in time is treated either by the leap-frog technique or by a modified Crank-Nicolson procedure, which generalizes the one suggested by Delfour, Fortin and Payne and possesses two useful conserved quantities.

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J. M. Sanz-Serna

Papers

Methods for the numerical solution of the nonlinear Schroedinger equation