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# Comparison of Numerical Schemes to Solve the Newton-Lorentz Equation

To better understand the transport of solar energetic particles, a conceptual understanding of the micro-physics of charged particle propagation in electric and magnetic fields is needed. The movement of charged particles are governed by the Newton-Lorentz equation, which becomes increasingly difficult to solve analytically for complicated electric and magnetic fields. In this work, the methods of Boris (1970) and Vay (2008), as well as a forth-order Runge-Kutta scheme, are investigated for numerically solving the Newton-Lorentz equation. This work focuses on accuracy rather than computational speed, since these numerical schemes will be used to analyse the motion of particles in turbulent electric and magnetic fields. The Larmor radius, deviation between the numerical and analytical position, as well as the execution time, are calculated and recorded for different time steps. It is found that the methods do not numerically change the energy of the particle, except the

Runge-Kutta method for large time steps; that the methods do not converge for large time steps and the accuracy is limited by the floating point accuracy for small time steps; that the Boris and Vay methods are second-order accurate, while the Runge-Kutta method is forth-order accurate; that the Runge-Kutta method is slightly slower than the other two methods and the evaluation of the tangent function in the Boris method does not add additional execution time, except for large time steps; and that a balance between accurate results and execution time suggest 16 steps per gyration for the Boris and Vay methods and 8 steps per gyration for the

Runge-Kutta method.

References:

Boris, J.P. 1970. Relativistic plasma simulation: optimization of a hybrid code. Numerical Simulation of Plasmas Conference Proceedings, 4, 3-67.

Vay, J.L. 2008. Simulation of beams or plasmas crossing at relativistic velocity. Physics of Plasmas, 15.