Implementation of high order spline interpolations for tracking charged particles in discretized fields

In numerical approaches, two main issues have to be addressed. First, an accurate solution for the time evolution of the particle position and velocity has to be computed. Second, an accurate interpolation of the fields e(r) and b(r), generally available only on a discrete grid, is needed. Although most of the information presented here is valid for a wide range of systems, the discussion is restricted to the simple case of particles submitted to electric and magnetic fields obtained from the magneto-hydrodynamic (MHD) equations: