Nonlinear parabolic equations are central to numerous applications in science and engineering, posing significant challenges for analytical solutions and necessitating efficient numerical methods. Exponential integrators have recently gained attention for handling stiff differential equations. This paper explores exponential Runge--Kutta methods for solving such equations, focusing on the simplified form $u^{\prime}(t)+A u(t)=B u(t)$, where $A$ generates an analytic semigroup and $B$ is relatively bounded with respect to $A$. By treating $A$ exactly and $B$ explicitly, we derive error bounds for exponential Runge--Kutta methods up to third order. Our analysis shows that these methods maintain their order under mild regularity conditions on the initial data $u_0$, while also addressing the phenomenon of order reduction in higher-order methods. Through a careful convergence analysis and numerical investigations, this study provides a comprehensive understanding of the applicability and limitations of exponential Runge--Kutta methods in solving linear parabolic equations involving two unbounded and non-commuting operators.

翻译：非线性抛物方程在科学与工程众多应用中处于核心地位，其解析求解面临显著挑战，因而需要高效的数值方法。近年来，指数积分器在处理刚性微分方程方面受到广泛关注。本文探讨用于求解此类方程的指数Runge--Kutta方法，重点关注简化形式$u^{\prime}(t)+A u(t)=B u(t)$，其中$A$生成解析半群，$B$相对于$A$相对有界。通过对$A$进行精确处理、对$B$进行显式处理，我们推导出最高三阶指数Runge--Kutta方法的误差界。分析表明，在初始数据$u_0$满足温和正则性条件下，这些方法能保持其收敛阶，同时解决了高阶方法中的阶数约化现象。通过细致的收敛性分析和数值实验，本研究全面阐释了指数Runge--Kutta方法在求解涉及两个无界非交换算子的线性抛物方程时的适用性与局限性。