TY - JOUR
T1 - Numerical Analysis of a Nonlinear Singularly Perturbed Delay Volterra Integro-Differential Equation on an Adaptive Grid
AU - Liu , Libin
AU - Chen , Yanping
AU - Liang , Ying
JO - Journal of Computational Mathematics
VL - 2
SP - 258
EP - 274
PY - 2022
DA - 2022/01
SN - 40
DO - http://doi.org/10.4208/jcm.2008-m2020-0063
UR - https://global-sci.org/intro/article_detail/jcm/20186.html
KW - Delay Volterra integro-differential equation, Singularly perturbed, Error analysis, Monitor function.
AB - <p style="text-align: justify;">In this paper, we study a nonlinear first-order singularly perturbed Volterra integro-differential equation with delay. This equation is discretized by the backward Euler for
differential part and the composite numerical quadrature formula for integral part for which
both an a priori and an a posteriori error analysis in the maximum norm are derived. Based
on the a priori error bound and mesh equidistribution principle, we prove that there exists
a mesh gives optimal first order convergence which is robust with respect to the perturbation parameter. The a posteriori error bound is used to choose a suitable monitor function
and design a corresponding adaptive grid generation algorithm. Furthermore, we extend
our presented adaptive grid algorithm to a class of second-order nonlinear singularly perturbed delay differential equations. Numerical results are provided to demonstrate the
effectiveness of our presented monitor function. Meanwhile, it is shown that the standard arc-length monitor function is unsuitable for this type of singularly perturbed delay
differential equations with a turning point.</p>