On the stability of second-order methods for DDEs and PDDEs
Abstract
In this paper, a one-parameter family of second-order implicit--explicit (IMEX) multistep methods is studied for delay differential equations (DDEs). The analysis is based on the matrix problem y'(t)=A y(t) + B y(t−τ(t)), where A is Hermitian negative definite but B is any matrix. For constant, grid-aligned delays, a field-of-values argument yields unconditional stability whenever a suitable generalized field of values Fp(A,B) is contained in the open unit disk. For a constant non-grid-aligned delay, we show that quadratic interpolation does not reduce the corresponding stability disk. For genuinely variable delays, the method is formulated as a nonautonomous augmented system, and product-level sufficient conditions are obtained from a common quadratic Lyapunov inequality and from a quantitative slow-variation hypothesis. The approach developed herein is illustrated through several examples in which the discussed theory is applied not only to DDEs, but also to semidiscrete parabolic problems given by partial delay differential equations with a diffusion term.
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BibTeX
@misc{rodriguezgutierrez2026stability,
title = {{On the stability of second-order methods for DDEs and PDDEs}},
author = {Javier Rodríguez-Gutiérrez and Jesús Martín-Vaquero},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33780},
eprint = {2610.33780},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33780},
}
RIS
TY - UNPB
TI - On the stability of second-order methods for DDEs and PDDEs
AU - Javier Rodríguez-Gutiérrez
AU - Jesús Martín-Vaquero
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33780
UR - https://arxiv.org/abs/2610.33780
AB - In this paper, a one-parameter family of second-order implicit--explicit (IMEX) multistep methods is studied for delay differential equations (DDEs). The analysis is based on the matrix problem y'(t)=A y(t) + B y(t−τ(t)), where A is Hermitian negative definite but B is any matrix. For constant, grid-aligned delays, a field-of-values argument yields unconditional stability whenever a suitable generalized field of values F_p(A,B) is contained in the open unit disk. For a constant non-grid-aligned delay, we show that quadratic interpolation does not reduce the corresponding stability disk. For genuinely variable delays, the method is formulated as a nonautonomous augmented system, and product-level sufficient conditions are obtained from a common quadratic Lyapunov inequality and from a quantitative slow-variation hypothesis. The approach developed herein is illustrated through several examples in which the discussed theory is applied not only to DDEs, but also to semidiscrete parabolic problems given by partial delay differential equations with a diffusion term.
ER -