Two-stage waveform relaxation method for the initial value problems with non-constant coefficients
- PDF / 387,770 Bytes
- 14 Pages / 439.37 x 666.142 pts Page_size
- 34 Downloads / 170 Views
Two-stage waveform relaxation method for the initial value problems with non-constant coefficients Zeinab Hassanzadeh · Davod Khojasteh Salkuyeh
Received: 3 August 2013 / Accepted: 4 October 2013 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013
Abstract In this paper, we present a two-stage waveform relaxation method applied to the initial value problems for the linear systems of ordinary differential equations in the form y (t) + A(t)y(t) = f (t). By making use of the forward Euler method, we derive sufficient conditions for the convergence of this method, when A(t) is M-matrix for every t ∈ [t0 , T ]. Finally some numerical experiments are given to illustrate some of the theoretical results. Keywords Two-stage · Waveform relaxation method · Euler method · Ordinary differential equations · Inner/outer · M-splitting Mathematics Subject Classification (2000)
34A09 · 34A12
1 Introduction Many scientific and engineering problems can be represented by linear systems of ordinary differential equations (ODEs) and differential-algebraic equations (DAEs). For instance, electrical networks, constrained mechanical systems of rigid bodies, control theory, singular perturbation and discretization of partial differential equations, etc. (see Brenan et al. 1989; Campbell 1980, 1982). Several iterative methods have been investigated to solve these kinds of problems. The two-stage iterative method was first proposed for solving systems of linear equations by Nichols (1973). After that, waveform relaxation (WR) methods have been developed in order to numerically solve systems of ODEs, hence two-stage waveform
Communicated by Ruben Spies. Z. Hassanzadeh Department of Mathematics, University of Mohaghegh Ardabili, P.O. Box. 179, Ardabil, Iran e-mail: [email protected] D. K. Salkuyeh (B) Faculty of Mathematical Sciences, University of Guilan, P.O. Box 1914, Rasht, Iran e-mail: [email protected]; [email protected]
123
Z. Hassanzadeh, D. K. Salkuyeh
relaxation (TSWR) method has been studied for ODEs and DAEs (see Garrappa 2004; Bao and Song 2011; Zhou and Huang 2008; Wang and Bai 2006). In recent years, for an initial value problem (IVP) of ODEs in the form y (t) + Ay(t) = f (t), y(t0 ) = y0 , t ∈ [t0 , T ], where A ∈ Cm×m is a nonsingular matrix and f (t) : [t0 , T ] −→ Cm is supposed continuous, the TSWR method has been proposed and investigated. For instance, the convergence analysis for A being an M-matrix is given in Garrappa (2004). In Wang and Bai (2006) the method is investigated when A is an H-matrix, and in Zhou and Huang (2008) the convergence analysis is restricted to Hermitian positive definite matrices. The main aim of this work is to extend the analysis of WR and TSWR methods to IVPs of systems of linear ODEs with non constant coefficients in the form y (t) + A(t)y(t) = f (t), (1) y(t0 ) = y0 , t ∈ [t0 , T ], where A(t) : [t0 , T ] −→ Rm×m is a nonsingular M-matrix for every t ∈ [t0 , T ] with continuous entries and f (t) : [t0 , T ] −→ Rm is supposed to be continu
Data Loading...
