Abstract

Various systems in nature have a Hamiltonian structure and therefore accurate time integrators for those systems are of great practical use. In this paper, a finite element method will be explored to derive symplectic time stepping schemes for (non-)autonomous systems in a systematic way. The technique used is a variational discontinuous Galerkin finite element method in time. This approach provides a unified framework to derive known and new symplectic time integrators. An extended analysis for the new time integrators will be provided. The analysis shows that a novel third order time integrator presented in this paper has excellent dispersion properties. These new time stepping schemes are necessary to get accurate and stable simulations of (forced) water waves and other non-autonomous variational systems, which we illustrate in our numerical results.

Highlights

  • The dynamics of various physical phenomena, such as the movement of pendulums, planets, or water waves can be described in a variational framework

  • In the construction of variational time integrators we choose to work with a discontinuous Galerkin finite element method in time, because it allows additional freedom to develop new symplectic integrators, for non-autonomous Hamiltonian systems

  • While our schemes are always variational by construction, the symplectic and stability conditions need to be verified. These conditions can be met depending on the choice of quadrature we use and the choice of the free parameter(s) in the jump conditions. This complements Zhao and Wei’s work [10], because we developed our stable, variational and symplectic integrators for applications to water wave problems, since these can be expressed as space and time discrete variational principles [9, 27]

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Summary

Introduction

The dynamics of various physical phenomena, such as the movement of pendulums, planets, or water waves can be described in a variational framework. In the construction of variational time integrators we choose to work with a discontinuous Galerkin finite element method in time, because it allows additional freedom to develop new symplectic integrators, for non-autonomous Hamiltonian systems. In this method the time domain is split into elements, and in each element the variables are approximated with polynomial expansions. Combining the discrete variational principle with the derived numerical flux, we obtain a unified approach to construct time integrators for (non)autonomous Hamiltonian systems. We have derived both well-known and novel symplectic time stepping schemes of first, second and third order accuracy. More details on the analysis of the time integration methods are presented in two Appendices

Dynamics of a Hamiltonian system with one degree of freedom
Discrete functional
Second order variational time discretization
New variational time integrators
Third order scheme
Linear stability
Symplecticity
Applications to non-autonomous systems
Numerical results
Nonlinear potential flow water waves
Conclusions
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