How to avoid order reduction in third-order exponential Runge--Kutta methods for problems with non-commutative operators?

TL;DR

The paper addresses order reduction in explicit exponential Runge–Kutta methods for stiff problems with non_commuting unbounded operators $A$ and $B$. It introduces a four_stage, third_order exponential RK method that preserves the intended order under strong order conditions, assuming higher regularity of the initial data, and provides a detailed analytic framework in the setting of analytic semigroups on a Banach space. Through a thorough error_recursion analysis and numerical experiments on a linear advection–diffusion model, the authors demonstrate that the proposed method avoids order reduction and achieves the expected accuracy, while a standard ETD RK method does not. The work yields rigorous convergence results, practical implementation guidance via augmented_matrix techniques, and suggests avenues for extending the approach to higher_order schemes.

Abstract

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.

Figures (2)

• Figure 1: The global error is shown as a function of the time step $\tau$.
• Figure 2: The global error is shown as a function of the time step $\tau$.

Theorems & Definitions (11)

• Lemma 3.1
• Lemma 3.2
• Remark 1
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Theorem 4.1
• proof
• Lemma 4.3