Instituto de Investigación
en Matemáticas

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New Composition Methods to Improve the Performance of Numerical Methods for Differential Equations

Sergio Blanes (Universidad Politécnica de Valencia)

Fecha: 01/12/2011 17:00
Lugar: Sala de Grados de la Facultad de Ciencias
Grupo: Departamento de Matemática Aplicada

In this talk I will present new techniques based on composition to raise the order of numerical methods for solving differential equations. This is illustrated for splitting methods, but it can be extended to other families of methods. Composition techniques mainly concern two steps: 1- To look for the order conditions to be satisfied by the parameters of the new method. 2- To solve these equations and to look for the solutions which provide the most efficient methods for different purposes. These two steps are frequently used to build, e.g. Geometric Integrators when basically either i) one has a symmetric method to be used as the basic method for composition, or the vector field is separable is solvable parts, and ii) only real solution for the coefficients are taken into account. We extend this analysis to I) complex solutions for the coefficients of the methods, which allow to use high order methods for some problems, like diffusion problems, which can not be integrated with splitting methods of order higher than two with real coefficients (these methods have shown good performance even on some problems where order reduction appear), and II) we extend the analysis to the case where either the basic method is not symmetric or the equation is separable in simpler parts but, which are not exactly solvable, and this requires to take into account additional order conditions.