Approximate preservation of quadratic first integrals by explicit Runge–Kutta methods |
| |
Authors: | M Calvo M P Laburta J I Montijano L Rández |
| |
Institution: | 1.Departamento de Matemática Aplicada,Universidad de Zaragoza,Zaragoza,Spain |
| |
Abstract: | 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. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|