Fig. 3

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 previous and present integrators. The circle, triangle, square, and x-mark show the results with Boris (1970), Umeda (2023), fourth-order Runge–Kutta integrators (RK4 to Eq.(1) directly), and the present study with the combination of RK4 and the tangent function, respectively