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：本研究首次构建了用于刚性抛物型偏微分方程时间积分的六阶指数龙格-库塔（ExpRK）方法。首先，我们利用指数B级数理论，基于仅包含必要树结构的集合，重新阐述了任意阶ExpRK方法的刚性阶条件。随后，我们明确给出了六阶方法所需的36个阶条件，并展示了收敛性结果。此外，我们成功求解了弱形式与强形式下的36个刚性阶条件，从而得到两个六阶并行级ExpRK格式系列。有趣的是，尽管这些新格式需要较高的级数，但其实现效率可与六阶段方法相媲美。数值实验验证了新格式的精度与高效性。