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.

翻译：本文基于试验方程$y'(t)=-A y(t) + B y(t-\tau)$研究了隐显式BDF方法对时滞微分方程（DDEs）的稳定性，其中$\tau$为常时滞，$A$为正定矩阵，而$B$可为任意矩阵。首先分析了矩阵可同时对角化的情形，但重点研究$A$和$B$不可同时对角化的情况。通过数值域的概念，证明了该方法无条件稳定的一个充分条件，以及另一个依赖于步长的稳定性条件。文中给出了多个数值算例，将所述理论应用于DDEs，同时也展示了在具有扩散项和时滞项的偏时滞微分方程所描述的抛物问题中的应用。