Electron. J. Math. Phys. Sci., 2002, 1,1, 14-46
STRANG-TYPE PRECONDITIONERS FOR SOLVING SYSTEM OF ODES BY BOUNDARY
VALUE METHODS
Raymond H. Chan 1*, Xiao-Qing Jin 2, Yue-Hung Tam 3
1 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong.
2 Faculty of Science and Technology, The University of Macau, Macau.
3 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
*Corresponding Author.
E-mail: rchan@math.cuhk.edu.hk
Received: 2 April
2002/ Accepted: 2 May 2002/ Published: 22 August 20002
Abstract: In
this paper, we survey some of the latest developments in using boundary value
methods for solving systems of ordinary differential equations with initial
values. These methods require the solutions of one or more nonsymmetric, large
and sparse linear systems. The GMRES method with the Strang-type preconditioner
is proposed for solving these linear systems. One of the main results is that
if an Aν1, ν2 stable boundary value
method is used for an m-by-m system of ODEs, then the
preconditioner is invertible and the preconditioned matrix can be decomposed as
1 + L where 1 is the identity matrix and the rank of L is
at most 2m(ν1+ν2) .It follows that when
the GMRES method is applied to solving the preconditioned systems, the method
will converge in at most 2m(ν1+ν2) + 1 iterations.
Applications to differential algebraic equations and delay differential
equations are also given.
Keywords: Boundary value method,
GMRES, ordinary differential equation, Strang-type preconditioner
AMS Mathematical Subject Classification: 65L05, 65L06, 65N10, 65N22
© 2002 by EJMAPS
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