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__Strong stability properties for some classes of nonlinear problems__

I. Higueras

**Abstract**

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.

**Bibliography**

[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.