Improving the accuracy of the discrete gradient method in the one-dimensional case

We present two numerical schemes of high accuracy for one-dimensional dynamical systems. They are modifications of the discrete gradient method and keep its advantages, including stability and conservation of the energy integral. However, their accuracy is higher by several orders of magnitude.

Figure 1

Relative error of the period for $\epsilon =0.02$ as a function of $p_0$. White triangles: leap frog; white diamonds: discrete gradient; black diamonds: modified discrete gradient $\left(\delta =\text{const}\right)$; black squares: locally exact discrete gradient; gray squares: symmetric modification of the locally exact discrete gradient.

Figure 2

Relative error of the period for $p_0=1.8$ as a function of $\epsilon$ (theoretical period $T_{th}=9.122196554$). Symbols: the same as in Fig. 1.

Figure 3

(Color online) $x_n$ as a function of $n$, very near the separatrix $\left(p_0=1.9999999999\right)$, $\epsilon =0.9$ for all gradient schemes, and $\epsilon =0.001$ for the leap-frog scheme. Symbols: the same as in Fig. 1 (black squares are hidden under gray squares). The solid line corresponds to the exact (continuous) solution (theoretical period $T_{th}=51.59687914$).