| Issue |
ESAIM: M2AN
Volume 60, Number 1, January-February 2026
|
|
|---|---|---|
| Page(s) | 223 - 246 | |
| DOI | https://doi.org/10.1051/m2an/2025103 | |
| Published online | 13 February 2026 | |
Convergence analysis and error estimates for the CSRK schemes to conserved gradient flows
1
College of Science, National University of Defense Technology, Changsha 410073, P.R. China
2
School of Mathematics and Statistics, and Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, P.R. China
* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
19
May
2025
Accepted:
21
December
2025
Abstract
We conduct a comprehensive convergence and error analysis of the second- and third-order unconditionally energy stable convex splitting Runge-Kutta (CSRK) methods for H−1 gradient flows with typical forms of free energy. Through the energy structure inherent to gradient flows, we are able to derive uniform-in-time bounds of the numerical solution in the H1, H2, and L6 norms. In turn, these functional bounds enable us to derive the associated estimates for the nonlinear error terms. Meanwhile, motivated by the fact that the diffusion coefficients are diagonally dominated in the CSRK numerical systems, the convergence results become available, based on a stage-by-stage analysis for the error evolutionary equations. The Cahn–Hilliard and phase-field crystal equations are two examples in the theoretical analysis. We also numerically compute some convergence results to validate the theorems proposed in this paper. This work deepens the theoretical foundation of CSRK methods and provides robust analytical tools for their application to conserved gradient flows.
Mathematics Subject Classification: 65M12 / 65G40 / 65L06 / 35G20
Key words: Convergence and error estimate / gradient flows / convex splitting Runge–Kutta / Cahn–Hilliard equation / phase-field crystal equation
© The authors. Published by EDP Sciences, SMAI 2026
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