Numerical Methods for y″ =f(x, y) via Rational Approximations for the Cosine

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Abstract
To investigate stability and phase lag, a numerical method is applied to the test equation y″ = −ω2y. Frequently, the characteristic equation of the resulting recurrence relation has the form ζ2− 2Rnm(v2)ζ + 1 = 0, where v = ωh, with h the steplength, and Rnm(v2) is a rational approximation for cos v. In this paper, properties of such approximations are used to provide a general framework for the study of stability intervals and orders of dispersion of a variety of one- and two-step methods. Upper bounds on the intervals of periodicity of explicit methods with maximum order of dispersion are established. It is shown that the order of dispersion of a P-stable method, for given n and m, cannot exceed 2m; a consequence is that, of the Padé approximants for cos v, only the [0/2m] approximants have modulus less than unity for all v2 >0. A complete characterization of P-stable methods of fourth order corresponding to the rational approximation R22(v2) is followed by several results for methods which have finite intervals of periodicity; in particular, we identify methods which have order of dispersion 6 or 8 with large intervals of periodicity. There is also a detailed discussion of P-stable methods of sixth order corresponding to the rational approximation R33(v2).

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