Continuous Extensions for Structural Runge–Kutta Methods
The so-called structural methods for systems of partitioned ordinary differential equations studied by Olemskoy are considered. An ODE system partitioning is based on special structure of right-hand side dependencies on the unknown functions. The methods are generalization of Runge–Kutta–Nyström methods and as the latter are more efficient than classical Runge–Kutta schemes for a wide range of systems. Polynomial interpolants for structural methods that can be used for dense output and in standard approach to solve delay differential equations are constructed. The proposed methods take fewer stages than the existing most general continuous Runge–Kutta methods. The orders of the constructed methods are checked with constant step integration of test delay differential equations. Also the global error to computational costs ratios are compared for new and known methods by solving the problems with variable time-step.
KeywordsContinuous methods Delay differential equations Runge–Kutta methods Structural partitioning
- 5.Eremin, A.S.: Modifikatsiya teorii pomechennykh dereviev dly stukturnogo metoda integrirovaniya sistem ODU (Labelled trees theory modification for structural method of solving ODE systems). Vestn. St-Petersburg Uni. (2), 15–21 (2009). (in Russian)Google Scholar
- 6.Eremin, A.S., Olemskoy, I.V.: Functional continuous Runge-Kutta methods for special systems. In: AIP Conference Proceedings, vol. 1738, p. 100003 (2016)Google Scholar
- 15.Olemskoy, I.V.: Modifikatsiya algoritma vydeleniya strukturnykh osobennostei (Modification of structural properties detection algorithm). Vestn. St-Petersburg Uni. (2), 55–64 (2006). (in Russian)Google Scholar
- 16.Olemskoy, I.V.: Metody Integrirovaniya System Strukturno Razdelyonnykh Differentsialnykh Uravnenii (Integration of Structurally Partitioned Systems of Ordinary Differential Equations). Saint-Petersburg State Univ., Saint-Petersburg (2009). (in Russian)Google Scholar
- 18.Paul, C.A.H.: A test set of functional differential equations. Technical report 243, Manchester Centre for Computational Mathematics, University of Manchester, February 1994Google Scholar