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Vietnam Journal of Mathematics 40:1 (2012)
115-126
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Stability of Two-Step-by-Two-Step IRK
Methods Based on Gauss-Legendre Collocation
Points and an Application
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Nguyen Huu Cong⋆⋆ and Nguyen Thu Thuy
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Faculty of Mathematics, Mechanics and
Informatics,
Vietnam National University, 334 Nguyen Trai,
Thanh Xuan, Hanoi, Vietnam
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Received October 6, 2011
Revised February 3, 2012
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Abstract. This paper investigates a class of IRK-type
methods for solving first-order stiff initial-value problems (IVPs). The
IRK-type methods are constructed by using coefficients of s-stage collocation Gauss-Legendre
IRK methods and other 2s-stage
collocation IRK methods. The collo-cation points used in the 2s-stage methods are chosen such
that at nth integration step,
their stage values can be used as the stage values of the associated
collocation Gauss-Legendre IRK methods for (n+2)th integration step. By this way we obtain the methods in
which the integration processes can be proceeded two-step-by-two-step. The
resulting IRK-type methods are called two-step-by-two-step IRK methods
based on Gauss-Legendre collocation points (TBTIRKG methods). Stability
considerations show that these TBTIRKG methods can be A-stable or $A(\alpha)$-stable which can be applied to stiff
IVPs with a fewer number of implicit relations that are to be solved in the
integr-ation process when compared with the traditional Gauss-Legendre IRK
methods. The stability inve-stigation results for TBTIRKG methods were
applied in considerations of the
asymptotic stability of a class of PC methods based on the TBTIRKG
methods.
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2000 Mathematics
Subject Classification. 65L06, 65L20.
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Keywords.
Implicit Runge-Kutta methods, stability.
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⋆⋆Current address: School of Graduate
Studies, Vietnam National University, 144 Xuan Thuy, Cau Giay, Hanoi,
Vietnam.
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Established
by Vietnam Academy of Science and Technology &
Vietnam Mathematical Society
Published by
Springer since January 2013
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