Selection strategies for set-valued Runge-Kutta methods. (English) Zbl 1118.65342

Baier, Robert

Li, Zhilin (ed.) et al., Numerical analysis and its applications. Third international conference, NAA 2004, Rousse, Bulgaria, June 29 – July 3, 2004. Revised selected papers. Berlin: Springer (ISBN 3-540-24937-0/pbk). Lecture Notes in Computer Science 3401, 149-157 (2005).

Summary: A general framework for proving an order of convergence for set-valued Runge Kutta methods is given in the case of linear differential inclusions, if the attainable set at a given time should be approximated. The set-valued method is interpreted as a (set-valued) quadrature method with disturbed values for the fundamental solution at the nodes of the quadrature method. If the precision of the quadrature method and the order of the disturbances fit together, then an overall order of convergence could be guaranteed. The results are applied to modified Euler method to emphasize the dependence on a suitable selection strategy (one strategy leads to an order breakdown).
For the entire collection see [Zbl 1069.65501].

MSC:

65L05	Numerical methods for initial value problems involving ordinary differential equations
65L06	Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34A30	Linear ordinary differential equations and systems

Keywords:

set-valued Runge-Kutta methods; linear differential inclusions; selection strategies; modified Euler