Numerical methods, energy conservation, and a new method for particle motion in magnetic fields

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This article mathematically examines the energy conservation of explicit Runge–Kutta methods and basic symplectic integration methods for a simple harmonic oscillator and a particle in a magnetic field. With the demonstration of the failure of these methods to conserve the energy of a particle in a magnetic field, a new integration method is presented for this type of system where energy conservation is dictated by the type of floating point variable used, not the size of the time step or the method order. Higher order versions of the method improve the accuracy in the calculation of the particle position and allow for more accurate calculations of the velocity in a spatially varying field. Furthermore, the new method is demonstrated to have improved phase accuracy when compared to several common implicit methods.