SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Contributed Talk

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Strong stability properties for some classes of nonlinear problems

I. Higueras

Spacial discretization of some partial differential problems (PDEs) give rise to ordinary differential equations (ODEs). Sometimes, the solutions to these PDEs have qualitative properties, e.g., monotonicity, positivity, etc., which are relevant in the context of the problem. In these cases, it is convenient to preserve these properties both in the spatial discretization of the PDE and in the time stepping process of the resulting ODE. A common class of methods widely used in the literature are Runge-Kutta methods. For these schemes, some of these qualitative properties can be ensured under certain stepsize restrictions given in terms of the Kraaijevanger's coefficient. However, for some problems, several schemes with trivial Kraaijevanger's coefficient also provide good numerical solutions. In this talk we will explain how, under additional conditions on the problem, some qualitative properties can be obtained for some methods with trivial Kraaijevanger's coefficient.

[1] R. Donat, I. Higueras and A. Martínez-Gavara, On stability issues for IMEX schemes applied to hyperbolic equations with stiff reaction terms, Math. Comput. 80 (2011), 2097-2126.
[2] I. Higueras, Strong Stability for Runge-Kutta Schemes on a Class of Nonlinear Problems, J. Sci. Comput., (to appear), DOI 10.1007/s10915-013-9715-y
[3] I. Higueras, Positivity propertirs for the classical fourth order Runge-Kutta method, Monografías de la Real Academia de Ciencias de Zaragoza, 33 (2010), 125-139.

Organized by         Universidad de Valladolid     IMUVA