Projection methods preserving Lyapunov functions

In this paper we consider ordinary differential equations with a known Lyapunov function. We study the use of Runge–Kutta methods provided with a dense output and a projection technique to preserve any given Lyapunov function. This approach extends previous work of Grimm and Quispel (BIT 45, 2005), allowing the use of Runge–Kutta methods for which the associated quadrature formula does not need to have positive or zero coefficients. Some numerical experiments show the good performance of the proposed technique.