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Numerical methods for \(y''=f(x,y)\) via rational approximations for the cosine. (English) Zbl 0675.65072

The paper gives a general framework for the study of periodicity intervals and orders of dispersion of a wide variety of methods for solving initial value problems of the form \((1)\quad y''=f(x,y),\) \(y(x_ 0)=y_ 0,\) \(y'(x_ 0)=z_ 0.\) The analysis is provided on the basis of the rational function \(R_{nm}(\nu^ 2)\) (n and m are degrees of numerator and denominator respectively) occurring in the characteristic equation \(\zeta^ 2-2R_{nm}(\nu^ 2)\zeta +1=0,\) where \(\nu =\omega h\) and h denotes the step size.
The present theorems establish (via the rational function \(R_{nm}(\nu^ 2))\) upper bounds on the interval of periodicity of explicit methods with maximum order of dispersion. Moreover, it is shown that the order of dispersion of a P-stable method cannot exceed 2m (for given n and m).
Reviewer: A.Marciniak

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
41A20 Approximation by rational functions
34A34 Nonlinear ordinary differential equations and systems
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