  The implicit compact finite-difference scheme was developed for evolutionary
partial differential parabolic and Schr\"odinger-type equations and systems
with a weak nonlinearity. To make a temporal step of the compact implicit
scheme we need to solve a non-linear system. We use for this step a simple
explicit difference scheme and then Newton -- Raphson iterations, which are
implemented by the double-sweep method. Numerical experiments confirm the 4-th
order of an algorithm. The Richardson extrapolation improves it up to the 6-th
order.
