JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 241984 10.1155/2012/241984 241984 Research Article On the Convergence of Continuous-Time Waveform Relaxation Methods for Singular Perturbation Initial Value Problems Zhao Yongxiang 1, 2 Li Li 1 Vigo-Aguiar Jesus 1 School of Mathematics and Statistics Chongqing Three Gorges University Wanzhou 404000 China sanxiau.edu.cn 2 Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan 411105 Xiangtan China xtu.edu.cn 2012 9 8 2012 2012 26 04 2012 24 06 2012 12 07 2012 2012 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.

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.

1. Introduction

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 ϵ multiplying the highest derivatives. Since the classical Lipschitz constant and one-sided Lipschitz constant are generally of size 𝒪(ϵ-1)  (0<ϵ1), 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. , 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  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: (1.1)xT=max0tTx, and the λ norm of exponential type: (1.2)xλ,T=max0tT{e-λtx}, where λ is any given positive number and · denotes an any given norm in Cn.

2. Convergence Analysis of the First Type 2.1. Linear SPPs

Consider the following linear singular perturbation initial value problem (2.1)ϵdy(t)dt+Ay(t)=b(t),t[0,T],  0<ϵ1,y(0)=y0, where the constant matrix ARn×n and the input function b(t)Rn, y0 is the given initial value, and ϵ is the singular perturbation parameter. The constant matrix A is split by A=A1-A2, then the system (2.1) can be written as (2.2)ϵdy(t)dt+A1y(t)=A2y(t)+b(t),t[0,T],  0<ϵ1,y(0)=y0, then we can obtain the following iterate scheme: (2.3)dy(k+1)(t)dt+1ϵA1y(k+1)(t)=1ϵA2y(k)(t)+b(t)ϵ,t[0,T],  0<ϵ1,y(k+1)(0)=y0,k=1,2,, here we can choose the initial iterative function y(0)(t)=y0, the above iteration is called a continuous-time waveform relaxation process.

For any fixed k0, from (2.3), we have, upon premultiplying by e(s-t)A1 and integrating from 0 to t, as following: (2.4)y(k+1)(t)=e-(A1t)/ϵy0+0te(s-t)A1/ϵ(A2ϵy(k)(s)+b(s)ϵ)ds. Let (2.5)(Ry)(t)=0te(s-t)A1/ϵA2ϵy(s)ds, then (2.5) can be written as (2.6)y(k+1)(t)=(Ryk)(t)+φ(t), where φ(t)=e-(A1t)/ϵy0+0te(s-t)A1/ϵ(b(s)/ϵ)ds. It is easy to see that is a Volterra convolution operator with the kernel function κ(t)=e-(A1t)/ϵ(A2/ϵ): (2.7)(Ry)(t)=κ(t)*y(t)=0tκ(s-t)y(s)ds, and is the waveform relaxation operator.

Theorem 2.1.

Let the waveform relaxation operator be defined in C([0,T],Rn). If the kernel function κ(t) is continuous in [0,T] and satisfies κTC, where C is a constant, then the sequence of functions y(k)(t) defined by (2.3) satisfy (2.8)y(k)(t)-y*(t)T(CT)kk!y(0)(t)-y*(t)T, where y*(t) is the exact solution of system (2.1).

Proof.

By the norm ·T in C([0,T]), we can obtain (2.9)RkT0Tκk(t)ds,k=1,2,, where κ1(t)=κ(t), κk(t)=κ(t)*κk-1(t).

In fact, from the given condition κTC, for any tT, we have (2.10)κk(t)C0tκk-1(t)ds, it can be obtained, by induction, that (2.11)κk(t)C(Ct)k-1(k-1)!, so (2.12)RkT(CT)kk!, which complete the proof.

Finally, we mention that the estimate (2.8) is superlinear convergence estimate, which reveals a rapid convergence behavior when k.

2.2. Nonlinear SPPs

Consider the following nonlinear singular perturbation initial value problem: (2.13)ϵdy(t)dt=f(y(t),t),t[0,T],  0<ϵ1,y(0)=y0, where y0 is the given initial value, ϵ is the singular perturbation parameter. f:Rn×[0,T]Rn is given continuous function mapping, and y(t) is unknown.

The continuous-time Waveform Relaxation algorithm for (2.13) is (2.14)dy(k+1)(t)dt=1ϵF(y(k+1)(t),y(k)(t),t),t[0,T],  0<ϵ1,y(k+1)(0)=y0,k=1,2,, where the splitting function F(u,v,t) determines the type of the Waveform Relaxation algorithm, and we assume that F(u,v,t) satisfy the following Lipschitz condition (2.15)F(u1,v1,t)-F(u2,v2,t)L1u1-u2+L2v1-v2.

By integrating the inequality (2.15) of both side from 0 to t, we have (2.16)y(k+1)(t)=y0+0t1ϵF(y(k+1)(s),y(k)(s),s)ds.

Let y^(t) denote the function that iterated by y(t) from one iteration step, like (2.16), denote y^(t)=(y)(t).

Theorem 2.2.

Assume that the splitting function F(u,v,t) in WR iteration process (2.14) is Lipschitz continuous with respect to u and v, then the continuous-time Waveform Relaxation algorithm (2.14) is convergent.

Proof.

We introduce another continuous function z(t), and denote z^(t)=(z)(t), then (2.17)y^(t)-z^(t)=1ϵ0t(F(y^(s),y(s),s)-F(z^(s),z(s),s))ds1ϵ0tF(y^(s),y(s),s)-F(z^(s),z(s),s)ds. Equations (2.15)–(2.17) yield (2.18)y^(t)-z^(t)L1ϵ0ty^(t)-z^(s)ds+L2ϵ0ty(s)-z(s)ds. From (2.18), we have, upon premultiplying by e-λt, the following: (2.19)e-λty^(t)-z^(t)L1ϵe-λt0ty^(s)-z^(s)ds+L2ϵe-λt0ty(s)-z(s)dsL1ϵe-λt0teλs(e-λsy^(s)-z^(s))ds+L2ϵe-λt0teλs(e-λsy(s)-z(s))ds(L1ϵmax0st{e-λsy^(s)-z^(s)}+L2ϵmax0st{e-λsy(s)-z(s)})e-λt0teλsds, because of e-λt0teλsds1/λ, and from the definition of λ norm of the exponential type, we have (2.20)y^(t)-z^(t)λ,TL1λϵy^(s)-z^(s)λ,T+L2λϵy(s)  -z(s)λ,T. It is easy to obtain (2.21)y^(t)-z^(t)λ,TL2λϵ-L1y(s)-z(s)λ,T. We can choose large enough λ such that γ=L2/(λϵ-L1)<1. 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 (3.1)x(t)+Ax(t)+By(t)=r(t),t[0,T],ϵdy(t)dt+Cx(t)+Dy(t)=s(t),0<ϵ1,x(0)=x0,y(0)=y0, where x0 and y0 are the given initial value, ϵ is the singular perturbation parameter, r(t)Rn1 and s(t)Rn2 are given functions. The constant matrices ARn1×n1, BRn1×n2, CRn2×n1, DRn2×n2 are split by A=A1-A2, B=B1-B2, C=C1-C2, D=D1-D2 respectively, x(t)Rn1 and y(t)Rn2 are unknowns. Then the system (3.1) can be written as (3.2)x(t)+A1x(t)+B1y(t)=A2x(t)+B2y(t)+r(t),t[0,T],ϵdy(t)dt+C1x(t)+D1y(t)=C2x(t)+D2y(t)+s(t),0<ϵ1,x(0)=x0,y(0)=y0. The continuous-time Waveform Relaxation algorithm for (3.1) is as follows: (3.3)dx(k+1)(t)dt+A1x(k+1)(t)+B1y(k+1)(t)=A2x(k)(t)+B2y(k)(t)+r(t),t[0,T],dy(k+1)(t)dt+C1ϵx(k+1)(t)+D1ϵy(k+1)(t)=C2ϵx(k)(t)+D2ϵx(k)(t)+1ϵs(t),0<ϵ1,x(k+1)(0)=x0,y(k+1)(0)=y0,k=1,2,, The matrix form of (3.3) reads (3.4)ddt(x(k+1)(t)y(k+1)(t))+(A1B1C1ϵD1ϵ)(x(k+1)(t)y(k+1)(t))=(A2B2C2ϵD2ϵ)(x(k)(t)y(k)(t))+(r(t)s(t)ϵ). Solve the equations (3.4), we can derive (3.5)(x(k+1)(t)y(k+1)(t))=exp(-(A1B1C1ϵD1ϵ)t)(x(k+1)(0)y(k+1)(0))+0texp((A1B1C1ϵD1ϵ)(s-t))((A2B2C2ϵD2ϵ)(x(k)(s)y(k)(s))+(r(s)s(s)ϵ))ds.

Denote εxk(t)=xk(t)-x(t), εyk(t)=yk(t)-y(t), where x(t) and y(t) are the exact solutions of (3.1). From (3.2) and (3.5), we can obtain (3.6)(εxk+1(t)εyk+1(t))=0texp((A1B1C1ϵD1ϵ)(s-t))(A2B2C2ϵD2ϵ)(εxk(s)εyk(s))ds, then (3.6) can be written as (3.7)(εxk+1(t)εyk+1(t))=(R(εxk(s)εyk(s)))(t), clearly, is a Volterra convolution operator with the kernel function (3.8)κ(t)=exp((-A1B1C1ϵD1ϵ)t)(A2B2C2ϵD2ϵ),(Ry)(t)=κ(t)*y(t)=0tκ(s-t)y(s)ds, is the Waveform Relaxation operator.

Theorem 3.1.

Let the waveform relaxation operator be defined in C([0,T],Rn). If the kernel function κ(t) is continuous in [0,T] and satisfies κTM, where M is a constant, then the sequence of functions (εxk+1(t),εyk+1(t))Tdefined by (3.6) satisfy (3.9)(εxk(t)εyk(t))Tkk!Mk(max0<s<Tεx0(s)max0<s<Tεy0(s)).

Proof.

Taking the norm in both side of (3.6) which reads (3.10)(εxk+1(t)εyk+1(t))0texp(A1B1C1ϵD1ϵ(s-t))A2B2C2ϵD2ϵ(εxk(s)εyk(s))ds. From the induction of (3.10) and the condition κTM, we can derive (3.11)(εx1(t)εy1(t))M(max0<s<Tεx0(s)max0<s<Tεy0(s))t. Moreover, we can derive (3.12)(εxk(t)εyk(t))Tkk!Mk(max0<s<Tεx0(s)max0<s<Tεy0(s)), thus, (εxk(t),εyk(t))T0 as k which complete the proof.

3.2. Nonlinear SPPs

Consider the following nonlinear singular perturbation initial value problem: (3.13)x(t)=f(x(t),y(t),t),t[0,T],ϵdy(t)dt=g(x(t),y(t),t),0<ϵ1,x(0)=x0,y(0)=y0, where x0 and y0 are given initial values, ϵ is the singular perturbation parameter, f:Rn1×Rn2×[0,T]Rn1 and g:Rn1×Rn2×[0,T]Rn2 are given continuous function mappings.

The continuous-time waveform relaxation algorithm for (3.13) is (3.14)x(k+1)(t)=F(x(k+1)(t),x(k)(t),y(k+1)(t),y(k)(t),t),t[0,T],dy(k+1)(t)dt=1ϵG(x(k+1)(t),x(k)(t),y(k+1)(t),y(k)(t),t),0<ϵ1,x(k+1)(0)=x0,y(k+1)(0)=y0,k=1,2,, where the splitting functions F(u1,u2,u3,u4,t) and G(u1,u2,u3,u4,t) determine the type of the waveform relaxation algorithm. we can derive from (3.14) that (3.15)x(k)(t)=F(x(k)(t),x(k-1)(t),y(k)(t),y(k-1)(t),t),t[0,T],dy(k)(t)dt=1ϵG(x(k)(t),x(k-1)(t),y(k)(t),y(k-1)(t),t),0<ϵ1,x(k)(0)=x0,y(k)(0)=y0,k=1,2,. Denote (3.16)εxk+1(t)=xk+1(t)-xk(t),εyk+1(t)=yk+1(t)-yk(t).

Theorem 3.2.

Assume that the matrices (F/ui) and (G/ui)  (i=1,2,3,4) of the splitting functions F(u1,u2,u3,u4,t) and G(u1,u2,u3,u4,t) are continuous, then the continuous-time waveform relaxation algorithm (3.14) is convergent.

Proof.

Subtracting (3.14) from (3.15), we have (3.17)dεxk+1(t)dt=Fu1εxk+1(t)+Fu2εxk(t)+Fu3εyk+1(t)+Fu4εyk(t),dεyk+1(t)dt=1ϵ(Gu1εxk+1(t)+Gu2εxk(t)+Gu3εyk+1(t)+Gu4εyk(t)), the matrix form of (3.17) reads (3.18)ddt(εxk+1(t)εyk+1(t))=(Fu1Fu31ϵGu11ϵGu3)(εxk+1(t)εyk+1(t))+(Fu2Fu41ϵGu21ϵGu4)(εxk(t)εyk(t)). Denote (3.19)εk(t)=(εxk(t),εyk(t))T,A(t)=(Fu1Fu31ϵGu11ϵGu3),B(t)=(Fu2Fu41ϵGu21ϵGu4). Then, we can derive (3.20)dεk+1(t)dt=A(t)εk+1(t)+B(t)εk(t). Assume that the basis matrix ϕ(t) satisfies (3.21)dϕ(t)dt=A(t)ϕ(t), then the solution of (3.20) can be written as (3.22)εk+1(t)=ϕ(t)0tϕ-1(s)B(s)εk(s)ds. From (3.22), we have, upon taking the norm in both side and premultiplying by e-λt, that (3.23)e-λtεk+1(t)=e-λt0tϕ(t)ϕ-1(s)B(s)εk(s)ds, furthermore, (3.24)max0tT{e-λtεk+1(t)}=max0tT{e-λt0tϕ(t)ϕ-1(s)B(s)εk(s)ds}, so (3.25)εk+1(t)λ,Tmax0tT{e-λt0tϕ(t)ϕ-1(s)B(s)eλse-λsεk(s)ds}max0tT{e-λt0tϕ(t)ϕ-1(s)B(s)eλse-λsεk(s)ds}Mεk(t)λ,Tmax0tT{e-λt0teλsds}Mλεk(t)λ,T, where M=max0tT,0st{ϕ(t)ϕ-1(s)B(s)}, and we can choose large enough λ such that M/λ<1, then the iterative error sequences {εk(t)} are convergent.

Acknowledgments

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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