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Preserving algebraic invariants with Runge-Kutta methods. (English) Zbl 0969.65069
The authors introduce a generalized class of Runge-Kutta schemes, namely Lie-group-type Runge-Kutta methods, designed to solve ordinary differential equations on Lie groups, and to preserve a large class of algebraic invariants. The conditions for the retention of algebraic invariants are imposed on the coefficients of the methods and a pair of partial differential equations that must be obeyed by the invariants. It is also proved that Runge-Kutta schemes cannot preserve any polynomial conservation law except for linear and quadratic.

MSC:
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A26 Geometric methods in ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations
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