A new compact finite difference scheme for solving the complex Ginzburg–Landau equation

The complex Ginzburg–Landau equation is often encountered in physics and engineering applications, such as nonlinear transmission lines, solitons, and superconductivity. However, it remains a challenge to develop simple, stable and accurate finite difference schemes for solving the equation because of the nonlinear term. Most of the existing schemes are obtained based on the Crank–Nicolson method, which is fully implicit and must be solved iteratively for each time step. In this article, we present a fourth-order accurate iterative scheme, which leads to a tri-diagonal linear system in 1D cases. We prove that the present scheme is unconditionally stable. The scheme is then extended to 2D cases. Numerical errors and convergence rates of the solutions are tested by several examples.