Analysis of the performance of numerical integration methods for the tracking of ultra-high energy cosmic rays

•The four methods currently examined have all presented great precision on calculating the particle's trajectory.•The most efficient method among all studied cases was the Boris one.•The New Euler method has exhibited the best energy conservation properties.•The Boris method has presented for all studied cases no growth tendency in the particle's energy with simulation time. We analyzed the performance of several numerical integration methods in the ultra-relativistic regime. The integration methods include the fourth order Runge-Kutta, Boris, Vay, and a new method (called New Euler). For a proton traveling in a circular trajectory through a uniform perpendicular magnetic field, we have calculated the relative percentage error of Larmor radius of the particle's trajectory, relative efficiency between the methods, and relative percentage error of the particle's energy as a function of CPU time. In these calculations, we have investigated the influence of the following parameters: particle's energy, magnetic field strength, and the integration step. On calculating the Larmor radius, the four methods examined have all presented great precision. Among all studied cases, the most efficient method was the Boris method, which completes within a given CPU time more than twice cycles (complete circles, or periods of revolution) given by the Runge-Kutta one. The least efficient was the New Euler method (0.08 cycles), followed by Vay one (0.21 cycles). On the calculation of the particle's energy, New Euler method has exhibited the best properties, with the maximum relative percentage error of the particle's energy of roughly 10−13% remaining constant during the entire simulation. Except in the cases with the biggest simulation step, when for the Runge-Kutta method the particle's energy diverges asymptotically from its initial value, and with the smallest simulation step and lowest energy, when the Boris method has presented less precision than the Runge-Kutta one. The Runge-Kutta and Boris methods have presented approximately the same precision, with the maximum relative percentage error of ∼10−11%. The Vay method has presented the worst precision, with the particle's energy diverging asymptotically for all studied cases, with the relative percentage error of the particle's energy reaching approximately 10−8%, in ten seconds of CPU time.