Fig. 1

Relative error \(\eta\) of the momentum vector \(\varvec{u}=(u_x,u_y)\) and position vector \(\varvec{r}=(x,y)\) from the theoretical solution as a function of \(\Delta t\) for the present integrators with the combination of RK4 and different number of Taylor series terms. The circle, triangle, square, and x-mark show the results with \(\Delta t\) term only, up to \(\Delta t^3\) terms, up to \(\Delta t^5\) terms, and tangent function, respectively