Energy-diminishing integration of gradient systems

For gradient systems in Euclidean space or on a Riemannian manifold the energy decreases monotonically along solutions. Algebraically stable Runge-Kutta methods are shown to also reduce the energy in each step under a mild step-size restriction. In particular, Radau IIA methods can combine energy monotonicity and damping in stiff gradient systems. Discrete-gradient methods and averaged vector field collocation methods are unconditionally energy-diminishing, but cannot achieve damping for very stiff gradient systems. The methods are discussed when they are applied to gradient systems in local coordinates as well as for manifolds given by constraints.