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.

Introduction

Neutral delay differential equations (NDDEs) has found applications in many areas of science (see, e.g., [1], [4], [8]). In the last several decades, a multitude of papers have been devoted to the linear stability of numerical methods (see, e.g., [5], [9], [11], [17], [19], [20], [30], [3]). Several researchers have investigated the stability of the solutions to nonlinear NDDEs with different forms in Hilbert space (see [2], [22], [23], [31], [24], [25], [26]), but little work has been published on the general variable delay problem and explicit and diagonally implicit Runge–Kutta methods, which have less computational amount per integration step than full implicit methods.

On the other hand, in order to surmount the restrict of inner product norm, Nevanlinna and Liniger [18] first, in 1979, considered ODEs test problem and studied nonlinear stability of one-leg methods applied to ODEs in Banach space. In 1983, Vanselow [21] researched stability of linear multistep methods for classes K1, K2λ^∗ and K3μ in Banach space. In 1987, Shoufu Li [12], [13] introduced test problem class K(α, λ^∗) of stiff ODEs in Banach space, obtained a series of stability results of numerical methods for test problem class K(α, λ^∗) (α ⩽ 0), including linear multistep methods and explicit and diagonally implicit Runge–Kutta methods, and proved that some numerical methods can preserve the contractivity of the systems (α ⩽ 0) under some conditions. Recently, Wen et al. [28], [29] have extended these results to test problem class D(α, β, λ^∗) of stiff delay differential equations (DDEs) and obtained some stability results of numerical methods (α ⩽ 0). Wang et al. [27] also studied the stability of θ-methods for nonlinear NDDEs with constant delays in Banach space.

The purpose of this paper is to investigate the stability and conditional contractivity of explicit and diagonally implicit Runge–Kutta methods for nonlinear NDDEs with variable delay in Banach space. We follow the approach designed by Shoufu Li for ODEs. In particular, Theorem 4.1 has its analogue in [14], with major changes in the proof. The main obstacle to our development is the numerical treatment of “neutral term”, resulting in direct evaluation and interpolation approximation for “neutral term”. This makes many of the arguments essentially different compared with the cases of ODEs and DDEs. In reward, we obtain some numerical results on stiff ODEs with α > 0 and stiff DDEs with α > 0.

The main results are proved in Section 4. The preceding sections give some basic concepts, including the problem classes D(α, β,γ, L, λ^∗), D_δ(α, β, γ, L, λ^∗) introduced in Section 2 and B₀-stability of explicit and diagonally implicit Runge–Kutta methods introduced in Section 3. The stability of the presented methods for problem class D_δ(α, β, γ, L, λ^∗) is discussed in Section 5. The final part of the paper gives some examples and two numerical experiments, which further confirm the main results.