Multi-step hybrid methods for special second-order differential equations y ″(t) = f(t,y(t))
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In this paper, multi-step hybrid methods for solving special second-order differential equations y ″(t) = f(t,y(t)) are presented and studied. The new methods inherit the frameworks of RKN methods and linear multi-step methods and include two-step hybrid methods proposed by Coleman (IMA J. Numer. Anal. 23, 197–220, 8) as special cases. The order conditions of the methods were derived by using the SN-series defined on the set SNT of SN-trees. Based on the order conditions, we construct two explicit four-step hybrid methods, which are convergent of order six and seven, respectively. Numerical results show that our new methods are more efficient in comparison with the well-known high quality methods proposed in the scientific literature.
KeywordsMulti-step hybrid methods Order conditions Simplifying conditions Explicit methods Special second-order differential equations
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