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In this paper, we consider the integration of the special second-order initial value problem. Hybrid Numerov methods are used, which are constructed in the sense of Runge-Kutta ones. Thus, the Taylor expansions at the internal points are matched properly in the final expression. A new family of such methods attaining eighth algebraic order is given at a cost of only 7 function evaluations per step. The new family provides us with an extra parameter, which is used to increase phase-lag order to 18. We proceed with numerical tests over a standard set of problems for these cases. Appendices implementing the symbolic construction of the methods and derivation of their coefficients are also given.
KEYWORDSconstant coefficients, hybrid Numerov methods, initial value problem, interval of periodicity, numerical solution, phase-lag
INTRODUCTIONWe consider the initial value problem of second-orderwhere f ∶ ℜ N+1 → ℜ N and y [0] , y ′[0] ∈ ℜ N . Equation 1 is of special form since the derivative y ′ is not included in f.Here, we will deal with numerical methods solving problem (1) especially when their solution is oscillatory. Many problems in theoretical physics and chemistry, in electronics, in celestial mechanics, quantum mechanical scattering theory, and many fields of engineering can be solved applying such methods. P-stability is a very useful property when dealing with periodic problems, 1,2 and implicit Numerov-type methods with off-step points were introduced by Hairer, 3 Cash, 4 and Chawla 5 especially for achieving this property.Phase-lag is another interesting property introduced by Brusa and Nigro. 6 In this case, the numerical solutions stay as close as possible to the angle of the theoretical solution of the simple harmonic oscillator, which serves as test problem. Explicit modifications of Numerov methods with reduced phase errors were constructed later by Chawla 7 and Chawla and Rao. [8][9][10] Math Meth Appl Sci. 2017;40:7867-7878.wileyonlinelibrary.com/journal/mma
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