This contribution is dedicated to the exploration of exponential operator splitting methods for the time integration of evolution equations. It entails the review of previous achievements as well as the depiction of novel results. The standard class of splitting methods involving real coefficients is contrasted with an alternative approach that relies on the incorporation of complex coefficients. In view of long-term computations for linear evolution equations, it is expedient to distinguish symmetric, symmetric-conjugate, and alternating-conjugate schemes. The scope of applications comprises high-order reaction-diffusion equations and complex Ginzburg-Landau equations, which are of relevance in the theories of patterns and superconductivity. Time-dependent Gross-Pitaevskii equations and their parabolic counterparts, which model the dynamics of Bose-Einstein condensates and arise in ground state computations, are formally included as special cases. Numerical experiments confirm the validity of theoretical stability conditions and global error bounds as well as the benefits of higher-order complex splitting methods in comparison with standard schemes.

翻译：本文致力于探索用于演化方程时间积分的指数算子分裂方法。内容包括回顾已有成果以及描述新的研究进展。我们将涉及实系数的标准分裂方法与另一种基于复系数的替代方法进行对比。针对线性演化方程的长期计算，区分对称格式、对称共轭格式以及交替共轭格式具有实际意义。应用范围涵盖高阶反应-扩散方程和复金兹堡-朗道方程，这些方程在图案形成理论和超导理论中具有重要意义。作为特例，形式上还包含了描述玻色-爱因斯坦凝聚体动力学及出现于基态计算中的含时格罗斯-皮塔耶夫斯基方程及其抛物对应方程。数值实验验证了理论稳定性条件和全局误差界的有效性，并证实了高阶复分裂方法相较于标准格式的优势。