Decoupling Runge-Kutta schemes for elliptic-parabolic problems
Abstract
We study the construction and convergence of semi-explicit and iterative decoupling schemes for an elliptic-parabolic problem using higher-order Runge-Kutta methods. For the semi-explicit schemes, which are constructed using a nearby delay system with time delays, we establish the convergence of th-order Runge-Kutta methods under a weak coupling condition. We develop the convergence analysis by adapting the Fourier stability and perturbation techniques of [Lubich, Ostermann, Math. Comp., 64(210):601--627, 1995]. The key tool is the generating function framework, in which the Runge-Kutta discretization is encoded through an operator-valued function. Stability estimates are then obtained via Parseval's identity on the unit circle. We further present convergence results for iterative (fixed-stress and undrained-split) higher-order Runge-Kutta schemes. Here, a spectral decomposition of the Schur complement operator is central. Finally, we provide numerical examples to verify the proven convergence results.
Cite
@article{arxiv.2605.22485,
title = {Decoupling Runge-Kutta schemes for elliptic-parabolic problems},
author = {Robert Altmann and Abdullah Mujahid and Benjamin Unger},
journal= {arXiv preprint arXiv:2605.22485},
year = {2026}
}