Abstract
This paper is concerned with the numerical stability of Runge-Kutta methods for a class of nonlinear functional differential and functional equations. The sufficient conditions for the stability and asymptotic stability of -algebraically stable Runge-Kutta methods are derived. A numerical test is given to confirm the theoretical results.
1. Introduction
This paper is concerned with the numerical solution of the following nonlinear functional differential and functional equations (FDFEs): with the initial conditions where is a real constant, and are unknown vectors of complex functions, and are given vectors of complex functions with appropriate domains of definition, and and are given vectors of complex functions which satisfy the consistency relation
Systems of the form (1) are sometimes called hybrid systems [1] or coupled delay differential and difference equations [2, 3]. They arise widely in the fields of science and technology, such as control systems, physics, and biology (see [1–6] and the references therein). In particular, they include neutral delay differential equations (NDDEs) as special cases. In fact, the explicit NDDEs are equivalent to while the implicit NDDEs are equivalent to
In recent years, numerical methods for explicit NDDEs and implicit NDDEs have been studied extensively and a significant number of numerical stability results have been found (see [7–20]). However, the above results of numerical stability cannot be applied to the more general problem (1). In 1999, Liu [6] discussed the numerical stability of Runge-Kutta collocation methods with a constrained grid and linear -methods with a uniformed grid for linear systems of FDFEs: where is a positive constant and , , , , , and are the coefficient matrices. Then, the asymptotic stability of linear multistep methods, one-leg methods, Runge-Kutta methods, multistep Runge-Kutta methods, and Rosenbrock methods for linear systems of FDFEs (8) was investigated in papers [21–23], respectively. Recently, Yu and Li [24] and Yu and Wen [25] dealt with the stability and asymptotic stability of the analytical and numerical solutions (obtained by one-leg methods) of nonlinear FDFEs (1), respectively. In the present paper, we further discuss the numerical stability of Runge-Kutta methods for the nonlinear FDFEs. The sufficient conditions for the stability and asymptotic stability of -algebraically stable Runge-Kutta methods are derived.
2. Stability of the Problem Class
Let be an inner product and the corresponding norm in complex -dimensional space ; assume that the mappings and in (1) satisfy the following conditions: where , , , , , and are real constants and .
Throughout this paper, we assume that the problem (1) has unique exact solution , and denote the problem class consisting of all problems (1) with (9)–(11) by class .
Remark 1. Inequality 2.1 means that we admit of stiffness of the problem, that is, admitting large value for the classical Lipschitz constant of with respect to the second argument (for the concept of stiffness we refer to [26, 27]).
Remark 2. Linear systems of FDFEs (8) belong to the class , where , is the logarithmic matrix norm corresponding to the inner product norm in , and , , , , .
For problems of the class , we have the following stability results (see [24]).
Theorem 3. Suppose the problem (1) belongs to the class and . Then one has the following inequalities:
Here and later, , denote the solution of any given perturbed problem of (1): with the initial conditions which satisfy the consistency relation
Theorem 4. Suppose the problem (1) belongs to the class and . Then one has which characterizes the asymptotic stability property of the problem (1).
3. Stability Analysis of Runge-Kutta Methods for FDFEs
An -stage Runge-Kutta method for ordinary differential equations (ODEs) can be expressed as where , , and . In this paper we always assume that () and .
The adaptation of the Runge-Kutta method (17) for solving the problem (1) leads to where the integration step size , is an arbitrarily given positive integer, , , , and denote approximations to , , and , respectively, and for , and is an approximation to which is obtained by using the following formula: where and for .
Similarly, applying the same method to the perturbed problem (13), we have where , , and denote approximations to , , and , respectively, and for , and is an approximation to which is obtained by using the following formula: where and for .
Definition 5 (see [28]). Let be real constants with . A Runge-Kutta method (17) is said to be -algebraically stable if there exists a diagonal nonnegative matrix such that is nonnegative definite, where and . Particularly, the -algebraically stable method is called algebraically stable for short.
Theorem 6. Assume that the Runge-Kutta method (17) is -algebraically stable with . Then the numerical solutions , and , , obtained by applying the corresponding method (18) to the problems (1) and (13) which belong to the class with , respectively, satisfy the global stability inequalities where depends only on the method, , , , , , and .
Proof. Let
and (), where denotes the integer part; then .
It follows from (18) and (20) that
Thus, it is easily obtained that (see [28])
where , , . In view of -algebraic stability of the method and , we get
By using conditions (9)–(11), we have
When , that is, , (30) leads to
On the other hand, when , that is, , using conditions (9)–(11) and , (30) leads to
Here and below, we define equal to 0 for . Combining (31) and (32) yields
Substituting (33) into (29) and using condition , we obtain
By induction, (34) gives
Therefore, there is a real constant depending only on the method, , , , , , and such that the inequality (23) holds. On the other hand, using condition (11) and , we have
and this completes the proof of Theorem 6.
Particularly, for the algebraically stable Runge-Kutta method, we have the following.
Corollary 7. Assume that the Runge-Kutta method (17) is algebraically stable. Then the numerical solutions , and , , obtained by applying the corresponding method (18) to the problems (1) and (13) which belong to the class with , respectively, satisfy the global stability inequalities where depends only on the method, , , , , , and .
Remark 8. It is well known that the formulae Gauss, Radau IA, Radau IIA, and Lobatto IIIC (for ODEs) are all algebraically stable. Therefore, in terms of Corollary 7, the corresponding methods are globally stable for solving the nonlinear FDFEs of the class which satisfy the condition .
In the following, we further discuss the asymptotic stability of the Runge-Kutta method.
Theorem 9. Assume that the Runge-Kutta method (17) is -algebraically stable with , (i.e., the matrix is positive definite), , and . Then the numerical solutions , and , , obtained by applying the corresponding method (18) to the problems (1) and (13) which belong to the class with , respectively, satisfy The relations (38) characterize the asymptotic stability property of the method.
Proof. Let ; thus . In terms of the proof of Theorem 6, we have By induction, (39) gives Since and , we easily obtain that On the other hand, while , denote ; thus (26) yields Substituting (42) into (27) leads to Noting that and (41), we easily obtain Furthermore, using condition (11), we have Considering that and (44), (45) leads to and this completes the proof of Theorem 9.
Particularly, for the algebraically stable Runge-Kutta method, we have the following.
Corollary 10. Assume that the Runge-Kutta method (17) is algebraically stable with , , and . Then the numerical solutions , and , , obtained by applying the corresponding method (18) to the problems (1) and (13) which belong to the class with , respectively, satisfy
Remark 11. It is well known that the formulae Radau IA, Radau IIA, and Lobatto IIIC (for ODEs) are algebraically stable with , and . Therefore, in terms of Corollary 10, the corresponding methods are asymptotically stable for solving the nonlinear FDFEs of the class which satisfy the condition .
Remark 12. In the paper [25], it is proved that an A-stable one-leg method is globally stable and a strongly A-stable one-leg method is asymptotically stable for FDFEs. However, any A-stable one-leg method has order at most two. In the present paper, the stability results are based on -algebraic stability of Runge-Kutta methods, which, in general, can be of high order.
4. Numerical Experiments
Consider the following initial value problem: After application of the numerical method of lines, we obtain the following FDFEs: where is the spatial step, is a natural number such that , , , and , . Then, the problem (49) belongs to the class with where the inner product is standard inner product. We take (i.e., ) for the numerical method of lines; thus the condition () is satisfied, which means the analytical solution of the problem (49) is stable and asymptotically stable.
As an example, we consider the 2-stage Radau IIA method: for solving the problem (49) and its perturbed problem, where the initial conditions of the perturbed problem are
According to the results of Corollaries 7 and 10, the corresponding method (for FDFEs) will be stable and asymptotically stable. We denote the numerical solutions of problem (49) and its perturbed problem , and , , respectively, where and are approximations to and , respectively. The values and are listed in Figure 1 (where the abscissa denotes variable ).
(a)
(b)
As a comparison, we consider the explicit 3-stage Runge-Kutta method for solving the problem (49) and its perturbed problem, and list the values and in Figure 2.
(a)
(b)
From Figure 1, one can see that the values and are bounded and tend to zero. This coincides with the results of Corollaries 7 and 10. However, for the explicit 3-stage Runge-Kutta method (53), which is not algebraically stable, the situation is inverse as one can see that the values are divergent as .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the referees for their many valuable suggestions. This work was supported by the NSF of China (11171282, 11371302) and the Scientific Research Project of the Science and Technology Department of Hunan Province (2013FJ4064).