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Approximate preservation of quadratic invariants by explicit Runge-Kutta methods

Calvo, M. ( ; Laburta Santamaría, María Pilar ( ; Montijano, Juan I. ( ; Rández, Luis (

Abstract: 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 for- mula with classical order four and seventh order of QI{conservation is obtained by choosing their coe±cients 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 °ow than standard fourth{ order methods at the price of two extra function evaluations per step. Furthermore, it is a practical and e±cient alternative to the standard implicit methods required for a complete preservation of QIs.

Language: eng
Department: Matemática Aplicada

Keyword(s): Runge-Kutta Methods ; Quadratic Invariants ; Pseudo-Symplectic Methods


 Record created 2009-05-13, last modified 2012-01-23

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