Order Reduction of Exponential Runge--Kutta Methods: Fourth-Order Schemes for Non-Commuting Operators
Abstract
This paper extends the convergence analysis of explicit exponential Runge--Kutta methods for linear parabolic problems u'(t) + Au(t) = Bu(t), where A generates an analytic semigroup and B is relatively bounded with respect to A, from the third-order case to fourth-order schemes. By establishing the global error recursion relation and extending the defect-based analytical framework, we identify the terms responsible for stiff order reduction when A and B do not commute. Numerical experiments are performed on a non-commuting advection-diffusion problem to validate the theoretical results. Numerical tests using the classical four-stage, fourth-order schemes of Krogstad and Strehmel \& Weiner exhibit an observed convergence order of approximately 2.75, which matches the theoretical prediction from the convergence analysis.
Related
- Linear bounds and sharp exponential growth for commuting matrix algebra
- Commuting Toeplitz operators with harmonic symbols on the Fock space
- Commuting analytic and co-analytic Toeplitz operators on the harmonic Bergman space
- Essentially semi-commuting singular integral operators with Cauchy kernel on L2
- Applications of the Projection Characterization and Symmetry Bootstrap for Operators that are Nearby Commuting Operators
BibTeX
@misc{dang2026order,
title = {{Order Reduction of Exponential Runge--Kutta Methods: Fourth-Order Schemes for Non-Commuting Operators}},
author = {Thi Tam Dang and Pablo Alexei Gazca-Orozco and Trung Hau Hoang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09932},
eprint = {2609.09932},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09932},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Order Reduction of Exponential Runge--Kutta Methods: Fourth-Order Schemes for Non-Commuting Operators
AU - Thi Tam Dang
AU - Pablo Alexei Gazca-Orozco
AU - Trung Hau Hoang
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09932
UR - https://arxiv.org/abs/2609.09932
AB - This paper extends the convergence analysis of explicit exponential Runge--Kutta methods for linear parabolic problems u'(t) + Au(t) = Bu(t), where A generates an analytic semigroup and B is relatively bounded with respect to A, from the third-order case to fourth-order schemes. By establishing the global error recursion relation and extending the defect-based analytical framework, we identify the terms responsible for stiff order reduction when A and B do not commute. Numerical experiments are performed on a non-commuting advection-diffusion problem to validate the theoretical results. Numerical tests using the classical four-stage, fourth-order schemes of Krogstad and Strehmel \& Weiner exhibit an observed convergence order of approximately 2.75, which matches the theoretical prediction from the convergence analysis.
ER -