On the stability of IMEX BDF methods for DDEs and PDDEs

In this paper, the stability of IMEX-BDF methods for delay differential equations (DDEs) is studied based on the test equation $y^{\prime }\left(t\right)=-Ay\left(t\right)+By(t-\tau )$, where $\tau$ is a constant delay, $A$ diagonalizes in $R^{+}$, but $B$ might be any matrix. First, it is analyzed the case where both matrices diagonalize simultaneously, and sufficient conditions to obtain linear stability are proved. However, the paper focuses on the case where the matrices $A$ and $B$ are not simultaneously diagonalizable. The concept of field of values, denoted by $F\left(\cdot \right)$, is used to prove a sufficient condition for unconditional stability of these methods: if $A$ is Hermitian, IMEX-BDF2 is unconditionally stable whenever $F\left(A^{-1}B\right)$ is contained in the disk centered at 0 and radius $\frac{1}{3}$, $D(0,\frac{1}{3})$, while IMEX-BDF3 is unconditionally stable whenever $F\left(A^{-1}B\right)\subseteq D(0,\frac{1}{7})$. Furthermore, sufficient conditions are derived to ensure stability, but according to the step size as a function of the radius of the disk containing $F\left(A^{-1}B\right)$. The approach developed herein is illustrated through several examples in which the discussed theory is applied not only to DDEs, but also to parabolic problems given by partial delay differential equations (PDDEs) with a diffusion term.