Approximate preservation of quadratic invariants by explicit Runge–Kutta methods

The construction of new explicit Runge–Kutta methods taking into account, not only their accuracy, but also the preservation of Quadratic Invariants (QIs) is studied. An expression of the error of conservation of a QI by a Runge–Kutta method is given, and a new six–stage formula with classical order four and seventh order of QI–conservation is obtained by choosing their coefficients so that they minimize both local and conservation errors. This formula, as well as other ones derived by Aubry and Chartier [1] and some standard formulas, have been tested with several problems with quadratic and general invariants. It is shown that the new fourth–order explicit method preserves much better the qualitative properties of the flow than standard fourth– order methods at the price of two extra function evaluations per step. Furthermore, it is a practical and efficient alternative to the standard implicit methods required for a complete preservation of QIs.