Approximate preservation of quadratic first integrals by explicit Runge-Kutta methods

The approximate preservation of quadratic first integrals (QFIs) of differential systems in the numerical integration with Runge-Kutta (RK) methods is studied. Conditions on the coefficients of the RK method to preserve all QFIs up to a given order are obtained, showing that the pseudo-symplectic methods studied by Aubry and Chartier (BIT 98(3):439-461, 1998) of algebraic order p preserve QFIs with order q = 2p. An expression of the error of conservation of QFIs by a RK method is given, and a new explicit six-stage formula with classical order four and seventh order of QFI-conservation is obtained by choosing their coefficients so that they minimize both local truncation and conservation errors. Several formulas with algebraic orders 3 and 4 and different orders of conservation have been tested with some problems with quadratic and general first integrals. It is shown that the new fourth-order explicit method preserves much better the qualitative properties of the flow than the standard fourth-order RK method at the price of two extra function evaluations per step and it is a practical and efficient alternative to the fully implicit methods required for a complete preservation of QFIs.