In this paper, the stability of $\theta$-methods for delay differential equations is studied based on the test equation $y'(t)=-A y(t) + B y(t-\tau)$, where $\tau$ is a constant delay and $A$ is a positive definite matrix. It is mainly considered the case where the matrices $A$ and $B$ are not simultaneosly diagonalizable and 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. The results obtained are also simplified for the case where the matrices $A$ and $B$ are simultaneously diagonalizable and compared with other similar works for the general case. Several numerical examples in which the theory discussed here is applied to parabolic problems given by partial delay differential equations with a diffusion term and a delayed term are presented, too.

翻译：本文基于测试方程 $y'(t)=-A y(t) + B y(t-\tau)$ 研究了 $\theta$-方法在延迟微分方程中的稳定性，其中 $\tau$ 为常延迟，$A$ 为正定矩阵。主要考虑矩阵 $A$ 与 $B$ 不可同时对角化的情形，利用数值域概念证明了这些方法无条件稳定性的一个充分条件，以及另一个保证其稳定性但与步长相关的条件。所得结果也简化为矩阵 $A$ 与 $B$ 可同时对角化的情形，并与一般情形下的其他类似工作进行了比较。此外，文中还给出了若干数值算例，将本文理论应用于由含扩散项和延迟项的偏延迟微分方程所描述的抛物型问题。