Research Article | Open Access
On the Convergence of Continuous-Time Waveform Relaxation Methods for Singular Perturbation Initial Value Problems
This paper extends the continuous-time waveform relaxation method to singular perturbation initial value problems. The sufficient conditions for convergence of continuous-time waveform relaxation methods for singular perturbation initial value problems are given.
Singular perturbation initial value problems play an important role in the research of various applied sciences, such as control theory, population dynamics, medical science, environment science, biology, and economics [1, 2]. These problems are characterized by a small parameter multiplying the highest derivatives. Since the classical Lipschitz constant and one-sided Lipschitz constant are generally of size , the classical convergence theory, B-convergence theory cannot be directly applied to singular perturbation initial value problems.
Waveform relaxation methods were introduced by Lelarasmee et al. . In recent years, these methods have been widely applied due to their flexibility, convenience, and efficiency. The problems of convergence of waveform relaxation methods for systems of ordinary differential equations, differential algebraic equations, integral equations, delay differential equations, and fractional differential equations were discussed by main authors (cf. [4–7], and references therein). In’t Hout  consider the convergence of waveform relaxation methods for stiff nonlinear ordinary differential equations. The monograph of Jiang  which entitled “Waveform Relaxation Methods” introduced these methods for various ODEs, differential algebraic equations, integral equations, delay differential equations and fractional differential equations, and PDEs. For more comprehensive survey on these methods and their applications, the reader is refered to the monograph in  and the references therein. Natesan et al., Vigo-Aguiar and Natesan [10–14] considers the general second order singular perturbation problems.
In this paper, we apply continuous waveform relaxation methods to single stiff singular perturbation initial value problems and obtain the corresponding convergence results.
In the rest parts of the text, we define the maximum norm as follows: and the norm of exponential type: where is any given positive number and denotes an any given norm in .
2. Convergence Analysis of the First Type
2.1. Linear SPPs
Consider the following linear singular perturbation initial value problem where the constant matrix and the input function , is the given initial value, and is the singular perturbation parameter. The constant matrix is split by , then the system (2.1) can be written as then we can obtain the following iterate scheme: here we can choose the initial iterative function , the above iteration is called a continuous-time waveform relaxation process.
For any fixed , from (2.3), we have, upon premultiplying by and integrating from 0 to , as following: Let then (2.5) can be written as where . It is easy to see that is a Volterra convolution operator with the kernel function : and is the waveform relaxation operator.
Theorem 2.1. Let the waveform relaxation operator be defined in . If the kernel function is continuous in and satisfies , where is a constant, then the sequence of functions defined by (2.3) satisfy where is the exact solution of system (2.1).
Proof. By the norm in , we can obtain
where , .
In fact, from the given condition , for any , we have it can be obtained, by induction, that so which complete the proof.
Finally, we mention that the estimate (2.8) is superlinear convergence estimate, which reveals a rapid convergence behavior when .
2.2. Nonlinear SPPs
Consider the following nonlinear singular perturbation initial value problem: where is the given initial value, is the singular perturbation parameter. is given continuous function mapping, and is unknown.
The continuous-time Waveform Relaxation algorithm for (2.13) is where the splitting function determines the type of the Waveform Relaxation algorithm, and we assume that satisfy the following Lipschitz condition
By integrating the inequality (2.15) of both side from 0 to , we have
Let denote the function that iterated by from one iteration step, like (2.16), denote .
Theorem 2.2. Assume that the splitting function in WR iteration process (2.14) is Lipschitz continuous with respect to and , then the continuous-time Waveform Relaxation algorithm (2.14) is convergent.
Proof. We introduce another continuous function , and denote , then Equations (2.15)–(2.17) yield From (2.18), we have, upon premultiplying by , the following: because of , and from the definition of norm of the exponential type, we have It is easy to obtain We can choose large enough such that . Thus, the Waveform Relaxation operator is a contractive operator under this norm. From the contractive mapping principle, we can derive that the continuous-time Waveform Relaxation algorithm (2.14) is convergent.
3. Convergence Analysis of The Second Type
3.1. Linear SPPs
Consider the following linear singular perturbation initial value problem where and are the given initial value, is the singular perturbation parameter, and are given functions. The constant matrices , , , are split by , , , respectively, and are unknowns. Then the system (3.1) can be written as The continuous-time Waveform Relaxation algorithm for (3.1) is as follows: The matrix form of (3.3) reads Solve the equations (3.4), we can derive
Denote , , where and are the exact solutions of (3.1). From (3.2) and (3.5), we can obtain then (3.6) can be written as clearly, is a Volterra convolution operator with the kernel function is the Waveform Relaxation operator.
Theorem 3.1. Let the waveform relaxation operator be defined in . If the kernel function is continuous in and satisfies , where is a constant, then the sequence of functions defined by (3.6) satisfy
3.2. Nonlinear SPPs
Consider the following nonlinear singular perturbation initial value problem: where and are given initial values, is the singular perturbation parameter, and are given continuous function mappings.
Theorem 3.2. Assume that the matrices and of the splitting functions and are continuous, then the continuous-time waveform relaxation algorithm (3.14) is convergent.
Proof. Subtracting (3.14) from (3.15), we have the matrix form of (3.17) reads Denote Then, we can derive Assume that the basis matrix satisfies then the solution of (3.20) can be written as From (3.22), we have, upon taking the norm in both side and premultiplying by , that furthermore, so where and we can choose large enough such that , then the iterative error sequences are convergent.
This work is supported by Projects from NSF of China (11126329 and 10971175), Specialized Research Fund for the Doctoral Program of Higher Education of China (20094301110001), NSF of Hunan Province (09JJ3002), and Projects from the Board of Education of Chongqing City (KJ121110).
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Copyright © 2012 Yongxiang Zhao and Li Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.