Order conditions for G-symplectic methods
General linear methods constitute a family of multistage and multivalue family of integration methods for ordinary differential equations that include both Runge-Kutta and multistep integrators. Although it has been shown that general linear methods cannot be symplectic unless they reduce to Runge-Kutta methods, they can possess the G-symplecticity property: a particular partitioned matrix involving the coefficients of the method, a real diagonal matrix and a non-singular matrix which is identically zero. This feature leads to the preservation of a more general quantity than in the case of symplectic Runge-Kutta methods. To implement these schemes it is necessary to start the procedure with a one-step method (the so-called starting method) and also taking care of the so-called parasitic behavior. In this paper, the order conditions of such a family of parasitism-free integrators are analyzed by using the B-series and rooted trees approach, both for the G-symplectic scheme and for the starting method. It is shown that the starting method greatly influences the conditions to be satisfied by the G-symplectic scheme to achieve a given order, and this influence is characterized in terms of trees. All order conditions up to order five are obtained and several methods of order four are proposed and illustrated on the gravitational two-body problem.
scientific article; zbMATH DE number 6537624