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

TL;DR

This paper introduces a compact implicit finite-difference scheme for weakly nonlinear parabolic and Schrödinger-type equations, achieving high-order accuracy through Richardson extrapolation and efficient nonlinear system solving.

Contribution

It develops a novel compact implicit scheme combined with explicit steps and Newton-Raphson iterations for improved accuracy and efficiency in solving nonlinear PDEs.

Findings

Numerical experiments confirm 4th order accuracy.

Richardson extrapolation increases accuracy to 6th order.

Efficient solution of nonlinear systems using double-sweep method.

Abstract

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.