Fig. 4

Relative error \(\eta\) of the momentum vector \(\varvec{u}=(u_x,u_y)\), the position vector \(\varvec{r}=(x,y)\), the constant for the elliptical trajectory C in Eq.(5), and the boosted Lorentz factor \(\gamma _B\) in Eq.(6) from the theoretical solution as a function of time for the previous and present integrators. (a) \(0 \le \omega _ct \le 100\) and (b) \(0 \le \omega _ct \le 10^7\). 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