General local energy-preserving integrators for solving multi-symplectic Hamiltonian PDEs

Abstract

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 Schrödinger equations and 2D nonlinear Schrödinger 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 \emph{long-term} behaviour of our new schemes.

References in corpus (3)

• Multisymplectic geometry, variational integrators, and nonlinear PDEs
• Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field" method
• Stability of exact solutions of the defocusing nonlinear Schrodinger equation with periodic potential in two dimensions

Cited by in corpus (7)

• Arbitrarily high-order structure-preserving schemes for the Gross-Pitaevskii equation with angular momentum rotation in three dimensions
• Energy preserving model order reduction of the nonlinear Schrödinger equation
• Global energy preserving model reduction for multi-symplectic PDEs
• Long-time momentum and actions behaviour of energy-preserving methods for semilinear wave equations via spatial spectral semi-discretizations
• A linearized and conservative Fourier pseudo-spectral method for the damped nonlinear Schrödinger equation in three dimensions
• Unconditional energy dissipation and error estimates of the SAV Fourier spectral method for nonlinear fractional generalized wave equation
• Mass- and energy-preserving exponential Runge-Kutta methods for the nonlinear Schrödinger equation