The Stability and Accuracy of The Adams-Bashforth-type Integrator

TL;DR

The paper analyzes an Adams-Bashforth-type integrator (ABTI) for time integration, challenging the notion of a uniform limiting stability as accuracy increases. By deriving the characteristic polynomial of the amplification matrix and employing root-locus/harmonic-analysis arguments, it shows Buvoli's conjecture does not hold and identifies a parabolic stability radius $|z|<1/e$ (with $ ext{Re}(z)<0$) that constrains achievable accuracy under a CFL-type condition. It explains an order-loss mechanism when the sampling parameter satisfies $s=q$, and provides a simple fix $s=q+1$ to restore the ideal order. The analysis extends to $L^2$-stability and error bounds for parabolic PDEs using a tensor-product framework and a CFL condition, with numerical verifications on ODE and PDE problems validating the theory. Overall, the work delivers practical stability- and accuracy-aware guidelines for high-order explicit time-stepping beyond classical Adams-Bashforth schemes.

Abstract

This paper presents stability and accuracy analysis of a high-order explicit time stepping scheme introduced by \cite[Section 2.2]{Buvoli2019}, which exhibits superior stability compared to classical Adams-Bashforth. A conjecture that is supported by several numerical phenomena in \cite[Figure 2.5]{Buvoli2018}, the method appears to remain stable when the accuracy approaches infinity, although it is not yet proven. It is regrettable that this hypothesis has been refuted from a fundamental perspective in harmonic analysis. Notwithstanding the aforementioned, this method displays considerably enhanced stability in comparison to conventional explicit schemes. Furthermore, we present a criterion for ascertaining the maximum permissible accuracy for a given specific parabolic stability radius. Conversely, the original method will lose one order associated with the expected accuracy, which can be recovered with a slight modification. Consequently, a unified analysis strategy for the \( L^2 \)-stability will be presented for extensional PDEs under the CFL condition. Finally, a selection of representative numerical examples will be shown in order to substantiate the theoretical analysis.

Figures (5)

• Figure 1: Obtain the initial solution vector $\mathbf{u}^{[0]}$ by iterator \ref{['eq:iterator']}(Left) and update each of component $u_j^{[n+1]}$ of solution vector $\mathbf{u}^{[n+1]}$ by propagator \ref{['eq:propagator']} (Right).
• Figure 2: The zeros of $p_{4}(z; \theta)$ on the $z$-plane (Left) and the simple transformation $\zeta(\theta) = - e^{-i \theta} z(\theta)$ on the $\zeta$-plane (Right). The zeros $\zeta(\theta)$ and $z(\theta)$ be marked by blue whenever $\theta \in [-\tfrac{\pi}{2}, \tfrac{\pi}{2}]$.
• Figure 3: The local plot of polynomial sequence $\tilde{p}_{n}(\zeta; \pi)$ (Left) and the graph of $\lvert \tilde{\mathcal{P}}(\varphi; \zeta) \rvert$ with $\varphi \in [0, 2 \pi]$ and $\zeta = - \tfrac{1}{e}, - \tfrac{1}{2e}, - \tfrac{1}{4e}$(Right)
• Figure 4: $\tau=r_2/4 \times h^2\times 0.9$ (Left), $\tau=r_2/4 \times h^2$ (Middle) and $\tau = r_2/4 \times h^2\times 1.1$ (Right).
• Figure 5: The stability domain described by the spectral radius of the original matrix $\mathbf{A} + z \mathbf{B}(\alpha)/\alpha = \mathbf{A} + z \mathbf{S}(\alpha)\mathbf{F}/\alpha$ (Left), the spectral radius of the equivalent matrix $\mathbf{e}_{1}\mathbf{e}_{1}^{\mathrm{T}} + z \mathbf{F}\mathbf{S}(\alpha)/\alpha$ (Middle) and the maximum module of the zeros of the derived characteristic polynomial (Right). The circle centered at the origin with a radius of $1/e$ marked with red line and the contour of stability region for $q = 2,\ldots, 16$ with blue one.

Theorems & Definitions (23)

• Definition 1.1
• Lemma 3.1
• proof
• Remark
• Theorem 3.2
• proof
• Remark
• Corollary 3.1
• Corollary 3.2
• Lemma 3.3