On the stability of IMEX BDF methods for DDEs and PDDEs

TL;DR

The paper develops a comprehensive stability analysis for IMEX-BDF methods applied to delay differential equations, focusing on the test problem $y'(t)=-A y(t)+B y(t-\tau)$ with $A$ positive definite and $B$ arbitrary. It first treats the simultaneous diagonalizability case and then extends to the general non-commuting case through a field-of-values (FOV) framework, deriving unconditional stability disks $D(0,1/3)$ and $D(0,1/7)$ for IMEX-BDF2 and IMEX-BDF3, respectively, and explicit step-size constraints via functions $\chi$ and $\tilde{\chi}$. The analysis leverages scalar stability results, eigenstructure, and FOV properties to bound $h$ in terms of spectral data or numerical ranges, and it is validated through linear and nonlinear PDDE simulations obtained via the method of lines. The results provide practical stability criteria for IMEX-BDF schemes in DDEs and PDDEs, with demonstrated applicability to parabolic problems and delayed Burgers-type equations. Overall, the work broadens the stability theory for IMEX multistep methods in delayed settings and suggests avenues for higher-order extensions and refined conditions for highly disparate spectra.

Abstract

In this paper, the stability of IMEX-BDF methods for delay differential equations (DDEs) is studied based on the test equation $y'(t)=-A y(t) + B y(t-τ)$, where $τ$ is a constant delay, $A$ is a positive definite matrix, but $B$ might be any matrix. First, it is analyzed the case where both matrices diagonalize simultaneously, but the paper focus in the case where the matrices $A$ and $B$ are not simultaneosly diagonalizable. The concept of field of values is used to prove a sufficient condition for unconditional stability of these methods and another condition which also guarantees their stability, but according to the step size. Several numerical examples in which the theory discussed here is applied to DDEs, but also parabolic problems given by partial delay differential equations with a diffusion term and a delayed term are presented.

Figures (10)

• Figure 1: $\Gamma_{z}$ with $m=0$, $u=0$, and $z=-1, \ldots,-50$ ($\Gamma_{-50}$ is the most interior, ..., $\Gamma_{-1}$ is the most exterior region.)
• Figure 2: $\Gamma_{-1}$ for different values of $m$.
• Figure 3: Representation of $\psi(z)$ and $\Tilde{\psi}(z)$ functions.
• Figure 4: Representation of $\psi'(z)$ and $\Tilde{\psi}'(z)$ functions.
• Figure 5: Location of the numerical range $F((-A)^{-1}B)$.

Theorems & Definitions (34)

• Definition 1
• Definition 2
• Theorem 3
• Theorem 4
• Definition 5
• Definition 6
• Definition 7
• Definition 8
• Definition 9
• Lemma 10