Nonlinear stability of explicit and diagonally implicit Runge–Kutta methods for neutral delay differential equations in Banach space

Abstract

This paper is devoted to the investigations into the nonlinear stability properties of explicit and diagonally implicit Runge–Kutta methods for solving neutral delay differential equations in Banach space. Two approaches to numerically treating “neutral term” are considered, which allow us to prove several results on numerical stability and conditional contractivity of explicit and diagonally implicit Runge–Kutta methods. In particular, some new results on stiff ODEs and stiff DDEs, which can be regarded as the generalizations of previous results in the literature, are obtained.