An efficient split-step compact finite difference method for cubic–quintic complex Ginzburg–Landau equations

We propose an efficient split-step compact finite difference method for the cubic–quintic complex Ginzburg–Landau (CQ CGL) equations both in one dimension and in multi-dimensions. The key point of this method is to separate the original CQ CGL equations into two nonlinear subproblems and one or several linear ones. The linear subproblems are solved by the compact finite difference schemes. As the nonlinear subproblems cannot be solved exactly, the Runge–Kutta method is applied and the total accuracy order is not reduced. The proposed method is convergent of second-order in time and fourth-order in space, which is confirmed numerically. Extensive numerical experiments are carried out to examine the performance of this method for the nonlinear Schrödinger equations, the cubic complex Ginzburg–Landau equation, and the CQ CGL equations. It is shown from all the numerical tests that the present method is efficient and reliable.