Exponential Runge–Kutta methods for delay differential equations
Introduction
Consider the delay differential equations (DDEs):where and is continuous, denotes a given positive constant, M is a real matrix, is required to be Lipschitz continuous with respect to on any bounded temporal interval , i.e., for any , and T is a given positive constant, is the corresponding Lipschitz constant. In fact, this system results from the semi-linearization of parabolic equations with delay term:
where , is a linear elliptic differential operator, f is a given function, and is a bounded domain in , and 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:
where for , is a real matrix, the components of provide approximations to the exact solutions and is an initial function corresponding to function . Furthermore, the vector is -dimensional, which is related to function and matrix .
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 with 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.
Section snippets
Convergence of exponential RK methods for DDEs
Based on variation-of-constants formula, the construction of exponential RK methods for DDEs (1) is outlined as follows.
Let h be a given constant step such that and be the corresponding grid points for , then
Applying the collocation method, then we have the numerical scheme:where
Stability analysis
In this section, we will discuss the stability of exponential RK methods for the following test problem:where for and is a rational function.
Here, we assume that all the eigenvalues of M have negative real parts, and M has linearly independent eigenvectors.
For the test Eq. (17), exponential RK methods (2), (3) read
where and
Numerical illustrations
Example 1 (Convergence). Consider the initial value problem (1) with
where , and for . Furthermore, the source function is chosen in such a way that the exact solution of this problem is just the same as the initial function.
Now, we apply 2-stage exponential RK method to this equation with collocation nodes and . By the recurrence relations (2), (3), we
Acknowledgements
The authors wish to thank the two anonymous referees for their valuable comments which helped us to improve the present paper. This paper was supported by Natural Science Foundation of Province and project HITC200710 of Science Research Foundation and Natural Scientific Research Innovation Foundation HIT.NSRIF.2009053 in Harbin Institute of Technology.
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