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An efficient optimized adaptive step-size hybrid block method for integrating differential systems. (English) Zbl 1433.65128

Summary: This paper deals with the development, analysis and implementation of an optimized hybrid block method having different features, for integrating numerically initial value ordinary differential systems. The hybrid nature of the proposed one-step scheme allows us to bypass the first Dahlquist’s barrier on linear multi-step methods. The theory of interpolation and collocation has been used in the development of the method. We assume an appropriate polynomial representation of the theoretical solution of the problem and consider three off-step points in a one-step block. One of these three off-step points is fixed and the other two off-step points are optimized in order to minimize the local truncation errors of the main method and other additional formula. The resulting scheme is of order five having the property of \(\mathcal{A}\)-stability. An embedded-type approach is used in order to formulate the proposed method in adaptive form, showing a high efficiency. The adaptive method is tested on well-known differential systems viz. the Robertson’s system, a Gear’s system, a system related with Jacobi elliptic functions, the Brusselator system, and the Van der Pol system, and compared with some well-known numerical codes in the scientific literature.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
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