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：本文首次构建了六阶指数龙格-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，基于仅包含一组必要树结构，重新表述了任意阶ExpRK方法的刚性阶条件。随后，明确给出了六阶方法所需的36个阶条件，并呈现了收敛性结果。此外，我们同时求解了弱形式和强形式下的36个刚性阶条件，由此衍生出两个系列的六阶并行阶段ExpRK格式。有趣的是，尽管这些新格式需要较多的阶段数，但其计算效率可媲美六阶段方法的成本。数值实验验证了新格式的精度与效率。