This paper investigates the performance of a subclass of exponential integrators, specifically explicit exponential Runge--Kutta methods. It is well known that third-order methods can suffer from order reduction when applied to linearized problems involving unbounded and non-commuting operators. In this work, we consider a fourth-stage third-order Runge--Kutta method, which successfully achieves the expected order of accuracy and avoids order reduction, as long as all required order conditions are satisfied. The convergence analysis is carried out under the assumption of higher regularity for the initial data. Numerical experiments are provided to validate the theoretical results.

翻译：本文研究了一类指数积分器——特别是显式指数Runge--Kutta方法的性能。众所周知，当应用于涉及无界且非交换算子的线性化问题时，三阶方法可能出现阶数缩减。在这项工作中，我们考虑了一种四阶段三阶Runge--Kutta方法，只要所有所需的阶条件得到满足，该方法就能成功达到预期的精度阶数并避免阶数缩减。收敛性分析是在初始数据具有更高正则性的假设下进行的。数值实验被提供以验证理论结果。