Symmetries of explicit Runge-Kutta methods
- 132 Downloads
A new (abstract algebraic) approach to the solution of the order conditions for Runge-Kutta methods (RK) and to the corresponding simplifying assumptions was suggested in Khashin (Can. Appl. Math. Q. 17(1), 555–569, 2009, Numer. Algorithm, 61(2), 1–11, 2012). The approach implied natural classification of the simplifying assumptions and allowed to find new RK methods of high orders. Here we further this approach. The new approach is based on the upper and lower Butcher’s algebras. Here we introduce auxiliary varieties ℳ D and prove that they are projective algebraic varieties (Theorem 3.2). In some cases they are completely described (Theorem 3.5). On the set of the 2-standard matrices (Definition 4.4) (RK methods with the property b 2 = 0) the one-dimensional symmetries are introduced. These symmetries allow to reduce consideration of the RK methods to the methods with c 2 = 2c 3/3, that is c 2can be removed from the list of unknowns. We formulate a hypothesis on how this method can be generalized to the case b 2 = b 3 = 0 where two-dimensional symmetries appear.
KeywordsRunge-Kutta Butcher system Butcher equations Butcher algebra Algebraic approach to Runge-Kutta
Unable to display preview. Download preview PDF.
- 1.Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley (2008)Google Scholar
- 4.Feagin, T.: A tenth-order Runge-Kutta method with error estimate. In: Proceedings of the IAENG Conference on Scientific Computing (2007)Google Scholar
- 5.Feagin, T.: High-Order Explicit Runge-Kutta Methods. http://sce.uhcl.edu/rungekutta/ (2013)
- 6.Hartshorne, R.: Algebraic Geometry. Springer-Verlag, New York (1993)Google Scholar
- 7.Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems, 2nd edn.Springer-Verlag (2000)Google Scholar
- 8.Jackiewicz, Z.: General Linear Methods for Ordinary Differential Equations. Wiley (2009)Google Scholar
- 10.Khashin, S.I.: Butcher algebras for Butcher systems. Numer. Algoritm. 61(2), 1–11 (2012)Google Scholar
- 11.Lang, S.: Algebra, Graduate Texts in Mathematics, 211, Revised 3rd edn. Springer-Verlag, New York (2002)Google Scholar
- 14.Verner, J.H.: A classification scheme for studying explicit Runge-Kutta pairs. In: Fatunla, S.O. (ed.) Scientific Computing pp. 201–225. (1994)Google Scholar
- 17.Verner, J.H.: Refuge for Runge-Kutta pairs. http://people.math.sfu.ca/~jverner/ (2013)