Symmetries of explicit Runge-Kutta methods

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 a"(3) (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 (2)can 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.