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arXiv · 2605.22485

Decoupling Runge-Kutta schemes for elliptic-parabolic problems

Robert Altmann·Abdullah Mujahid·Benjamin Unger
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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 $k$ time delays, we establish the convergence of $k$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.

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Robert Altmann, Abdullah Mujahid, Benjamin Unger. 2026-05-21. Decoupling Runge-Kutta schemes for elliptic-parabolic problems. https://arxiv.org/abs/2605.22485

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