This work constructs the first-ever sixth-order exponential Runge--Kutta (ExpRK) methods for the time integration of stiff parabolic PDEs. First, we leverage the exponential B-series theory to restate the stiff order conditions for ExpRK methods of arbitrary order based on an essential set of trees only. Then, we explicitly provide the 36 order conditions required for sixth-order methods and present convergence results. In addition, we are able to solve the 36 stiff order conditions in both their weak and strong forms, resulting in two families of sixth-order parallel stages ExpRK schemes. Interestingly, while these new schemes require a high number of stages, they can be implemented efficiently similar to the cost of a 6-stage method. Numerical experiments are given to confirm the accuracy and efficiency of the new schemes.

翻译：本文首次构造了六阶指数龙格-库塔（ExpRK）方法，用于刚性抛物型偏微分方程的时间积分。首先，我们利用指数B级数理论，基于仅包含必要树的集合，重新表述了任意阶ExpRK方法的刚性阶条件。然后，我们明确给出了六阶方法所需的36个阶条件，并给出了收敛性结果。此外，我们能够同时求解弱形式和强形式的36个刚性阶条件，从而得到两类六阶并行阶段ExpRK方案。有趣的是，尽管这些新方案需要较高的阶段数，但其实现效率可与六阶段方法相当。数值实验验证了新方案的精度和效率。