P-stable Hybrid Super-implicit Methods for Periodic Initial Value Problems
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Authors
M. Mehdizadeh Khalsaraei
- Faculty of Mathematical Science, University of Maragheh, Maragheh, Iran.
M. Molayi
- Faculty of Mathematical Science, University of Maragheh, Maragheh, Iran.
Abstract
This paper deals with a class of symmetric (hybrid) P-stable methods for the numerical solution of
special second order initial value problems (IVPs). For linear multistep methods, Lambert and Watson
[5], had shown that a P-stable method is necessarily implicit and that the maximum order attainable by a
P -stable method is at most two. P-stability is important in the case of ’periodic stiffness’ as it is termed
by Lambert and Watson [5], that is, when the solution consists of an oscillation of moderate frequency
with a high frequency oscillation of small amplitude superimposed. In order to overcome the order-barrier
on linear multistep P-stable methods, we developed a new type of implicit formulas of linear multistep
methods. The formulas, which we call to be hybrid super-implicit, are of more implicitness than the socalled
implicit formulas in the sense that they require the knowledge of functions not only at the past and
present time-step but also at the future ones. In the cases when the right hand side of IVP is very complex,
the super-implicit methods are preferred. Also, we have used off-step points which allow us to derive Pstable
schemes of high order. We report numerical experiments to illustrate the accuracy and
implementation aspects of this class of methods.
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ISRP Style
M. Mehdizadeh Khalsaraei, M. Molayi, P-stable Hybrid Super-implicit Methods for Periodic Initial Value Problems, Journal of Mathematics and Computer Science, 15 (2015), no. 2, 129-136
AMA Style
Khalsaraei M. Mehdizadeh, Molayi M., P-stable Hybrid Super-implicit Methods for Periodic Initial Value Problems. J Math Comput SCI-JM. (2015); 15(2):129-136
Chicago/Turabian Style
Khalsaraei, M. Mehdizadeh, Molayi, M.. "P-stable Hybrid Super-implicit Methods for Periodic Initial Value Problems." Journal of Mathematics and Computer Science, 15, no. 2 (2015): 129-136
Keywords
- Initial value problems
- Super-implicit
- Hybrid methods
- Off-step points
- P-stability.
MSC
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