In this paper, we consider the task of efficiently computing the numerical solution of evolutionary complex Ginzburg--Landau equations. To this aim, we employ high-order exponential methods of splitting and Lawson type for the time integration. These schemes enjoy favorable stability properties and, in particular, do not show restrictions on the time step size due to the underlying stiffness of the models. The needed actions of matrix exponentials are efficiently realized with pointwise operations in Fourier space (when the model is considered with periodic boundary conditions) or by using a tensor-oriented approach that suitably employs the so-called $\mu$-mode products (when the semidiscretization in space is performed with finite differences). The overall effectiveness of the approach is demonstrated by running simulations on a variety of two- and three-dimensional (systems of) complex Ginzburg--Landau equations with cubic and cubic-quintic nonlinearities, which are widely considered in literature to model relevant physical phenomena. In fact, in all instances high-order exponential-type schemes can outperform standard techniques to integrate in time the models under consideration, i.e., the well-known split-step method and the explicit fourth-order Runge--Kutta integrator.

翻译：本文探讨了高效计算演化型复金兹堡-朗道方程数值解的任务。为此，我们采用分裂型与Lawson型的高阶指数方法进行时间积分。这些格式具有优越的稳定性特性，尤其不会因模型本身的刚性而对时间步长产生限制。矩阵指数所需的运算通过傅里叶空间中的逐点操作（当模型采用周期边界条件时）或采用张量导向方法（当空间半离散化采用有限差分时）高效实现，该方法适当地运用了所谓的μ-模乘积。通过对具有三次和三次-五次非线性的二维及三维（系统）复金兹堡-朗道方程进行数值模拟，我们验证了该方法的整体有效性——这些方程在文献中被广泛用于建模相关物理现象。事实上，在所有实例中，高阶指数型格式相较于所考虑模型的标准时间积分技术（即著名的分步法和显式四阶龙格-库塔积分器）均展现出更优性能。