Premium
Strang‐type preconditioners applied to ordinary and neutral differential‐algebraic equations
Author(s) -
Zhang Chengjian,
Chen Hao,
Wang Leiming
Publication year - 2011
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.770
Subject(s) - generalized minimal residual method , mathematics , discretization , ordinary differential equation , differential algebraic equation , algebraic equation , convergence (economics) , boundary value problem , algebraic number , backward differentiation formula , linear system , differential equation , mathematical analysis , nonlinear system , physics , quantum mechanics , economic growth , economics
This paper deals with boundary‐value methods (BVMs) for ordinary and neutral differential‐algebraic equations. Different from what has been done in Lei and Jin ( Lecture Notes in Computer Science , vol. 1988. Springer: Berlin, 2001; 505–512), here, we directly use BVMs to discretize the equations. The discretization will lead to a nonsymmetric large‐sparse linear system, which can be solved by the GMRES method. In order to accelerate the convergence rate of GMRES method, two Strang‐type block‐circulant preconditioners are suggested: one is for ordinary differential‐algebraic equations (ODAEs), and the other is for neutral differential‐algebraic equations (NDAEs). Under some suitable conditions, it is shown that the preconditioners are invertible, the spectra of the preconditioned systems are clustered, and the solution of iteration converges very rapidly. The numerical experiments further illustrate the effectiveness of the methods. Copyright © 2011 John Wiley & Sons, Ltd.