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A one-step integration routine for normal differential systems, based on Gauss-Legendre quadrature. (English) Zbl 0683.65051

Collocation methods at Gaussian points for the numerical solution of ordinary differential equations are considered. It is well known [cf. E. Hairer, S. P. Nørsett and G. Wanner: Solving ordinary differential equations. I: Nonstiff problems (1987; Zbl 0638.65058), pp. 206] that a collocation method at s Gaussian points is equivalent to an implicit s-stages Runge-Kutta method of order 2s, and these methods have been extensively studied. Therefore the proposed methods can be considered as reformulations of well known Runge-Kutta methods. The authors have implemented these collocation methods for \(s\leq 4\) and applied them to find bifurcation points of periodic solutions of the Duffing equation.
Concerning the implementation, the implicit equations which arise at each step are solved by functional iteration and the starting values are computed by extrapolation of an Hermite interpolation of the solution in the previous step. On the other hand, the local error is controlled by comparison of the higher order coefficients of the truncated Legendre series of the solution in two consecutive steps.
Reviewer: M.Calvo

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems

Citations:

Zbl 0638.65058
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References:

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