TAILIEUCHUNG - Báo cáo " Fully parallel methods for a class of linear partial differential-algebraic equations "

This note deals with two fully parallel methods for solving linear partial differentialalgebraic equations (PDAEs) of the form: Aut + B∆u = f(x, t) where A is a singular, symmetric and nonnegative matrix, while B is a symmetric positive define matrix. The stability and convergence of proposed methods are discussed. Some numerical experiments on high-performance computers are also reported. | VNU Journal of Science Mathematics - Physics 23 2007 201-209 Fully parallel methods for a class of linear partial differential-algebraic equations Vu Tien Dung Department of Mathematics Mechanics Informatics College of Science VNU 334 Nguyen Trai Thanh Xuan Hanoi Vietnam Received 30 November 2007 received in revised form 12 December 2007 Abstract. This note deals with two fully parallel methods for solving linear partial differential-algebraic equations PDAEs of the form Aut BAu f x t 1 where A is a singular symmetric and nonnegative matrix while B is a symmetric positive define matrix. The stability and convergence of proposed methods are discussed. Some numerical experiments on high-performance computers are also reported. Keywords Differential-algebraic equation DAE partial differential-algebraic equation PDAE nonnegative pencil of matrices parallel method 1. Introduction Recently there has been a growing interest in the analysis and numerical solution of PDAEs because of their importance in various applications such as plasma physics magneto hydro dynamics electrical mechanical and chemical engineering etc. Although the numerical solution for differential-algebraic equations DAEs and PDAEs has been studied intensively 1 2 until now we have not found any results on parallel methods for PDAEs. This problem will be studied here for a special case. The paper is organized as follows. Section 2 deals with some properties of the so called nonnegative pencils of matrices. In Section 3 we describe two parallel methods for solving linear PDAEs whose coefficients found a nonnegative pencil of matrices. The solvability and convergence of these methods are studied. Finally in section 4 some numerical examples are discussed. 2. Properties of nonnegative pencils of matrices In what follows we will consider a pencil of matrices A B where A G Rraxra is a singular symmetric and nonnegative matrix with rank A r n and B G Rraxra is a symmetric positive define matrix. Such a pencil

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