Compact difference schemes for weakly-nonlinear parabolic and Schrodinger-type equations and systems

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.