Decoupling Runge-Kutta schemes for elliptic-parabolic problems

TL;DR

This paper develops and analyzes higher-order Runge-Kutta decoupling schemes for elliptic-parabolic problems, establishing convergence under weak coupling conditions and verifying results with numerical examples.

Contribution

It introduces a novel framework using generating functions for convergence analysis of decoupling schemes with higher-order Runge-Kutta methods.

Findings

Convergence of kth-order Runge-Kutta methods under weak coupling.

Stability estimates obtained via Fourier and Parseval techniques.

Numerical examples confirm theoretical convergence results.

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.