Geometric Derivation and Analysis of Multi-Symplectic Numerical Schemes for Differential Equations

Abstract

In the current work we present a class of numerical techniques for the solution of multi-symplectic PDEs arising at various physical problems. We first consider the advantages of discrete variational principles and how to use them in order to create multi-symplectic integrators. We then consider the nonstandard finite difference framework from which these integrators derive. The latter is now expressed at the appropriate discrete jet bundle, using triangle and square discretization. The preservation of the discrete multi-symplectic structure by the numerical schemes is shown for several one- and two-dimensional test cases, like the linear wave equation and the nonlinear Klein–Gordon equation.

Appendix

By denoting sinc$(\xi )=\sin {}(\xi )/\xi$, special cases of the exponential integrators described using (10) can be obtained, i.e.,

• Gautschi type exponential integrators [36] for

  $$\displaystyle \begin{aligned} \psi(\varOmega h)=\text{sinc}^2\left(\frac{\varOmega h}{2}\right), \qquad \phi(\varOmega h)=1 \end{aligned}$$

• Deuflhard type exponential integrators [37] for

  $$\displaystyle \begin{aligned} \psi(\varOmega h)=\text{sinc}(\varOmega h), \qquad \phi(\varOmega h)=1 \end{aligned}$$

• García-Archilla et al. type exponential integrators [38] for

  $$\displaystyle \begin{aligned} \psi(\varOmega h)=\text{sinc}^2(\varOmega h), \qquad \phi(\varOmega h)=\text{sinc}(\varOmega h) \end{aligned}$$

Finally, in [1] a way to write the Störmer–Verlet algorithm as an exponential integrators is presented.