Stability analysis of a singlerate and multirate predictor-corrector scheme for overlapping grids

TL;DR

This paper conducts a matrix stability analysis of predictor-corrector schemes for overlapping grids, revealing novel odd-even stability behaviors and proposing modifications to improve stability for solving incompressible Navier-Stokes equations.

Contribution

It introduces a detailed stability analysis of singlerate and multirate predictor-corrector schemes, uncovering odd-even stability differences and proposing modifications for enhanced stability.

Findings

High-order PC scheme stability increases with resolution and overlap.

Odd and even corrector iterations exhibit distinct stability behaviors.

Modified schemes show monotonic stability increase with corrector iterations.

Abstract

We use matrix stability analysis for a singlerate and multirate predictor-corrector scheme (PC) used to solve the incompressible Navier-Stokes equations (INSE) in overlapping grids. By simplifying the stability analysis with the unsteady heat equation in 1D, we demonstrate that, as expected, the stability of the PC scheme increases with increase in the resolution and overlap of subdomains. For singlerate timestepping, we also find that the high-order PC scheme is stable when the number of corrector iterations ($Q$) is odd. This difference in the stability of odd- and even-$Q$ is novel and has not been demonstrated in the literature for overlapping grid-based methods. We address the odd-even behavior in the stability of the PC scheme by modifying the last corrector iterate, which leads to a scheme whose stability increases monotonically with $Q$. For multirate timestepping, we observe that the stability of the PC scheme depends on the timestep ratio ($\eta$). For $\eta=2$, even-$Q$ is more stable than odd-$Q$. For $\eta\ge3$, even-$Q$ is more stable than odd-$Q$ for a small nondimensional timestep size and the odd-even behavior vanishes as the timestep size increases. The stability analysis presented in this work gives novel insight into a high-order temporal discretization for ODEs and PDEs, and has helped us develop an improved PC scheme for solving the incompressible Navier-Stokes equations.