Positivity properties for the classical fourth order Runge-Kutta method
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Autores: Inmaculada Higueras Sanz
- Localización: Monografías de la Real Academia de Ciencias Exactas, Físicas, Químicas y Naturales de Zaragoza, ISSN 1132-6360, Nº. 33, 2010, págs. 125-139
- Idioma: inglés
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Resumen
Over the last few years a great effort has been done to develop Runge-Kutta (RK) methods that preserve properties such as monotonicity or contractivity for convex funtionals, or positivity. Provided that these properties hold for the explicit Euler scheme under certain stepsize restriction, it has been proved that these properties can also be maintained by some higher order RK methods under a modified stepsize.
As this restriction includes the radius of absolute monotonicity of the RK scheme, strictly positive radius are required in order to obtain the desired properties with non trivial stepsizes. However, at least from the numerical positivity point of view, some authors have reported fairly good numerical results for some RK methods with zero radius, e.g. the classical fourth order four stages RK scheme. In this paper, we analyze this method and prove that, for some class of problems, it also preserves positivity. The study done strongly relies on the concept of region of absolute monotonicity for additive RK methods.
