Abstract
We derive a new class of linear multistep methods (LMMs) via the interpolation and collocation technique. We discuss the use of these methods as boundary value methods and block unification methods for the numerical approximation of the general second-order initial and boundary value problems. The convergence of these families of methods is also established. Several test problems are given to show a computational comparison of these methods in terms of accuracy and the computational efficiency.
Highlights
Linear multistep methods (LMMs) are widely used for the numerical integration of ordinary differential equations
It has been shown in [11] that symmetric schemes are the best candidates to be used as final methods
We have proposed a main method and additional methods which are obtained from the same continuous scheme
Summary
Linear multistep methods (LMMs) are widely used for the numerical integration of ordinary differential equations. The IVMs are used for the numerical integration of initial value problems [1,2,3,4] If these additional conditions are specified as initial and final conditions (or methods) so that they form a discrete analog of the continuous boundary value problems, we have the boundary value methods (BVMs). The BVMs have been used for the numerical integration of first-order initial and boundary value problems and their convergence and stability analysis have been fully discussed [5,6,7,8,9,10]. Biala and Jator [11] developed BVMs for the direct solution of the general second-order initial and boundary value problems arising from the semidiscretization of three-dimensional partial differential equations.
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