Exponential Runge–Kutta methods for delay differential equations

Abstract

This paper deals with convergence and stability of exponential Runge–Kutta methods of collocation type for delay differential equations. It is proved that these kinds of numerical methods converge at least with their stage order. Moreover, a sufficient condition of the numerical stability is provided. Finally, some numerical examples are presented to illustrate the main conclusions.

Introduction

Consider the delay differential equations (DDEs):

$$u^{\prime }(t)=Mu(t)+g(t\text{,}u(t)\text{,}u(t-\tau ))\text{,}t>0\text{,}$$

where $u(t)=\varphi (t)$ and $\varphi :[-\tau \text{,}0]\to R^d$ is continuous, $\tau$ denotes a given positive constant, M is a real $d\times d$ matrix, $g(t\text{,}u\text{,}v):R\times R^d\times R^d\to R^d$ is required to be Lipschitz continuous with respect to $u\text{,}v$ on any bounded temporal interval $[0\text{,}T]$, i.e., $\left|\right|g(t\text{,}u\text{,}v)-g(t\text{,}û\text{,}\hat{v})\left|\right|\le L(\left|\right|u-û\left|\right|+\left|\right|v-\hat{v}\left|\right|)$ for any $u\text{,}û\text{,}v\text{,}\hat{v}\in R^d$, $t\in [0\text{,}T]$ and T is a given positive constant, $L$ is the corresponding Lipschitz constant. In fact, this system results from the semi-linearization of parabolic equations with delay term:

$$\begin{array}{c}\frac{\partial }{\partial t}v(x\text{,}t)=Lv(x\text{,}t)+f(t\text{,}v(x\text{,}t)\text{,}v(x\text{,}t-\tau ))\text{,}t>0\text{,}\hfill \\ v(x\text{,}t)=v_0(x\text{,}t)\text{,}t\in [-\tau \text{,}0]\text{,}x\in \Omega \text{,}\hfill \\ v(x\text{,}t)=g(x\text{,}t)\text{,}t>0\text{,}x\in \partial \Omega \text{,}\hfill \end{array}$$

where $\tau >0$, $L$ is a linear elliptic differential operator, f is a given function, $x\in \Omega$ and $\Omega$ is a bounded domain in $R^p$, $v_0$ and $g$ are given initial and boundary functions, respectively. Applying the process of semi-discretization with respect to the spatial variable x, we can obtain the system:

$$V^{\prime }(t)=BV(t)+F(t\text{,}V(t)\text{,}V(t-\tau ))+R(t)\text{,}t>0\text{,}$$

where $V(t)=V_0(t)$ for $t\in [-\tau \text{,}0]$, $B$ is a real $N\times N$ matrix, the components of $V(t)\in R^N$ provide approximations to the exact solutions and $V_0$ is an initial function corresponding to function $v_0$. Furthermore, the vector $R(t)$ is $N$-dimensional, which is related to function $g$ and matrix $B$.

To solve DDEs numerically, a vast class of step-by-step methods, such as Runge–Kutta (RK) methods and linear multi-step methods, has been discussed. Many papers were devoted to convergence and stability analysis in last decades, see [1]. Recently, some papers were concerned with exponential RK methods. The convergence of exponential RK methods was considered for linear and semi-linear parabolic problems in [3]. These numerical methods were also performed for the time integration of semi-linear parabolic problems in [4]. The unconditional stability of exponential RK methods were studied for semi-linear systems of ordinary differential equations (ODEs) with a stiff linear part and a non-stiff nonlinear part in [8]. And the numerical stability of linear delay parabolic equations was discussed by applying step-by-step methods to the resulting DDEs in [10]. However, little is known about the convergence and stability of exponential RK methods for DDEs (1) so far.

In this paper, the convergence of exponential RK methods is studied on any bounded interval $[0\text{,}T]$ with $T/\tau \in Z^{+}$ and an error estimate is derived under some assumptions. Moreover, the asymptotic stability of exponential RK methods for a particular class of test problems is also analyzed.