In this paper, we consider the task of efficiently computing the numerical solution of evolutionary complex Ginzburg--Landau equations on Cartesian product domains with homogeneous Dirichlet/Neumann or periodic boundary conditions. To this aim, we employ for the time integration high-order exponential methods of splitting and Lawson type with constant time step size. 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 by using a tensor-oriented approach that suitably employs the so-called $\mu$-mode product (when the semidiscretization in space is performed with finite differences) or with pointwise operations in Fourier space (when the model is considered with periodic boundary conditions). 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 or cubic-quintic nonlinearities, which are widely considered in literature to model relevant physical phenomena. In fact, we show that high-order exponential-type schemes may outperform standard techniques to integrate in time the models under consideration, i.e., the well-known second-order split-step method and the explicit fourth-order Runge--Kutta integrator, for stringent accuracies.

翻译：本文研究在满足齐次狄利克雷/诺伊曼边界条件或周期边界条件的笛卡尔积域上，高效计算演化型复杂金兹堡-朗道方程数值解的问题。为实现这一目标，我们采用具有恒定时间步长的高阶指数型时间积分方法，包括分裂格式和Lawson型格式。这些格式具备良好的稳定性，尤其不会因模型固有的刚性而导致时间步长受限。矩阵指数所需运算通过张量导向方法高效实现：当空间采用有限差分进行半离散时，该方法恰当地运用所谓的$\mu$-模积；当模型考虑周期边界条件时，则通过傅里叶空间中的逐点运算完成。通过对多种具有立方或立方-五次非线性项的二维及三维（方程组）复杂金兹堡-朗道方程进行仿真计算，验证了本方法的整体有效性——这些方程在文献中被广泛用于模拟相关物理现象。实验表明，在严格精度要求下，高阶指数型格式在积分所研究模型时，能够超越经典的时间积分技术，即著名的二阶分裂步方法和显式四阶龙格-库塔积分器。