This research project is framed in the field of analysis, development and implementation of numerical methods for solving systems of differential equations. It is intended that the methods under consideration allow to build software in line with current requirements, to be adressed to specific problems of practical interest. Specifically, we propose five research lines splitted into two projects: One of them developed by members of the University of La Laguna and the other by the ones of the University of Zaragoza, although a close personal and scientific relationship exists between the two groups.

In the first subproject we propose two lines of research. The first one focuses on the implementation techniques for SAFERK methods with the objective of developing a code efficient integration with variable order and step and apply it to solve stiff problems, singular perturbation and differential-algebraic equations with high precision. The second line is intended to the time integration of differential equations arising in the semidiscretization in the spatial variables of partial differential equations. In particular, we investigate the methods of type W-AMF with order greater or equal to two (in PDE sense) paying attention to the order reduction due to boundary conditions in parabolic problems. It will also explore peer methods combined with Krylov and AMF techniques as well as exponential peer methods for this class of problems.

In the second subproject we propose three lines of research. The first one is devoted to the design of two-step peer methods, with the following goals: Find optimal linearly implicit A-stable methods for solving stiff problems; Find functionally fitted peer methods with absolute stability regions specially designed for certain types of spectra suitable for problems with stiff or of oscillatory type. The second line is devoted to the study of embedded pairs of low storage Runge-Kutta methods for differential problems of high dimension and its extension to the functionally fitted case. In the last line we address the development of Runge-Kutta schemes that inherit the properties of energy change in dissipative problems and new conservative integration methods suitable for highly oscillatory problems .