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Stability by order stars for non-linear theta-methods based on means. (English) Zbl 1182.65129
Summary: The linear stability analysis of non-linear one-step methods based on means is studied by means of the concept of stability regions and order stars. Concretely, non-linear \(\theta\)-methods based on harmonic, contraharmonic, quadratic, geometric, Heronian, centroidal and logarithmic means are considered. Their stability diagrams and order stars show their \(A\)-stability for \(\theta \geq 1/2\), and \(L\)-stability in some cases. Order stars in the Riemann surface are a requirement for non-linear one-step methods. The advantages and disadvantages of this technique are presented.

MSC:
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L05 Numerical methods for initial value problems
34A34 Nonlinear ordinary differential equations and systems, general theory
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