Compact difference schemes for weakly-nonlinear parabolic and Schrodinger-type equations and systems
Provides a high-order numerical method for solving a class of nonlinear PDEs, but the approach is incremental (combining existing techniques).
The authors developed a compact finite-difference scheme for weakly-nonlinear parabolic and Schrödinger-type equations, achieving 4th-order accuracy, improved to 6th-order via Richardson extrapolation.
The implicit compact finite-difference scheme was developed for evolutionary partial differential parabolic and Schrödinger-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.