Error Estimates of Some Splitting Schemes for Charged-Particle Dynamics under Strong Magnetic Field

In this work, we consider the error estimates of some splitting schemes for the charged-particle dynamics under a strong magnetic field. We first propose a novel energy-preserving splitting scheme with computational cost per step independent of the strength of the magnetic field. Then under the maximal ordering scaling case, we establish for the scheme and in fact for a class of Lie--Trotter-type splitting schemes a uniform (in the strength of the magnetic field) and optimal error bound in the position and in the velocity parallel to the magnetic field. For the general strong magnetic field case, the modulated Fourier expansions of the exact and the numerical solutions are constructed to obtain a favorable dependence of the error on the strength of the magnetic field. Numerical experiments are presented to illustrate the error and energy behavior of the splitting schemes.