[P-EM17] Space Plasma Physics: Theory and Simulation

[PEM17-P02] Higher-order integrators for relativistic equations of motion for charged particles.

*Riku Ozaki¹, Takayuki Umeda¹, Yoshizumi Miyoshi¹ (1.Institute for Space-Environmental Research, Nagoya University)

The purpose of this study is to improve the accuracy of a numerical method for integrating the relativistic equation of motion for charged particles. The Runge-Kutta integrator (RK4) is a classical method for numerically solving various difference equations. Although RK4 has a fourth-order accuracy in time, RK4 does not satisfy the conservation laws in the relativistic motion of charged particles. The Boris integrator [Boris,1970], which has the second-order accuracy in time, has been used conventionally due to the following reasons. The Boris integrator is time-reversible, is simple for implementing to numerical code, and satisfies the energy conservation law during the gyration of charged particles. However, it has been known that the Boris integrator does not give the exact relativistic motion of charged particles for a large Lorentz factor. Recently, a new integrator has been developed based on the theoretical solution to the relativistic equation of motion for charged particles, which is called as the Umeda integrator [Umeda, 2023]. The Umeda integrator is an improved version of the Boris integrator, which satisfies several conservation laws and gives the exact relativistic E-cross-B drift velocity. However, the Umeda integrator has the second-order accuracy in time and is less accurate than RK4. In the present study, we improve the accuracy of the Umeda integrator with multi-step techniques. First, the Umeda integrator is combined with the Euler method, the midpoint rule, the trapezoidal rule, the Heun method, and RK4. Second, the Taylor series expansion of the relativistic gyration angle is performed, and higher-degree terms are included. It is demonstrated that the order of the numerical accuracy with the Euler method, the midpoint rule, and the trapezoidal rule is second-order in time. However, the order of the numerical accuracy with the Heun method and RK4 are improved against the Umeda integrator.