Order Reduction of Exponential Runge--Kutta Methods: Non-Commuting Operators

TL;DR

This work analyzes exponential Runge--Kutta methods for linear parabolic problems of the form $u'(t)+A u(t)=B u(t)$, where $A$ generates an analytic semigroup and $B$ is relatively bounded with respect to $A$. By treating $A$ exactly and $B$ explicitly, the authors derive rigorous error bounds up to third order within an analytic-semigroup framework, showing that first- and second-order schemes preserve their convergence order under mild data regularity, while third-order schemes exhibit order reduction to $2.5$ in a general Banach-space setting (with potential improvements to about $2.75$ in $L^2$ under specific schemes and regularity). The convergence analysis hinges on precise defect representations, error recursions, and intricate bounds for unbounded non-commuting operators, complemented by numerical experiments on a 1D advection–diffusion model that confirm the predicted rates. Collectively, the results clarify the applicability and limitations of exponential Runge--Kutta methods for linear parabolic equations with non-commuting unbounded operators, guiding both theory and computational practice.

Abstract

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.

Figures (2)

• Figure 1: The global error is shown as a function of the time step $\tau$ for the different methods. '1st-order' refers to the first-order method, with similar interpretations for higher-order methods.
• Figure 2: The global error is shown as a function of the time step $\tau$ for the method given in \ref{['eq23']}.

Theorems & Definitions (28)

• Lemma 2.1
• Definition 2.1
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Theorem 4.1
• proof
• Remark 4.1
• Lemma 4.3