On the stability of IMEX BDF methods for DDEs and PDDEs

TL;DR

This paper investigates the stability of IMEX-BDF methods for delay differential equations and partial delay differential equations, providing new stability conditions and demonstrating their application through numerical examples.

Contribution

It introduces new stability criteria for IMEX-BDF methods when matrices are not simultaneously diagonalizable, expanding understanding of their stability for DDEs and PDDEs.

Findings

Derived sufficient conditions for unconditional stability.

Established step size-dependent stability conditions.

Validated theory with numerical examples on DDEs and PDDEs.

Abstract

In this paper, the stability of IMEX-BDF methods for delay differential equations (DDEs) is studied based on the test equation $y'(t)=-A y(t) + B y(t-\tau)$, where $\tau$ 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.