Energy-diminishing integration of gradient systems

Abstract

For gradient systems in Euclidean space or on a Riemannian manifold the energy decreases monotoni-cally along solutions. Algebraically stable Runge–Kutta methods are shown to also reduce the energyin each step under a mild step size restriction. In particular, Radau IIA methods can combine energymonotonicity and damping in stiff gradient systems. Discrete-gradient methods and averaged vector fieldcollocation methods are unconditionally energy-diminishing, but cannot achieve damping for very stiffgradient systems. The methods are discussed when they are applied to gradient systems in local coordi-nates as well as for manifolds given by constraints.Keywords: Gradient flow, energy dissipation, implicit Runge–Kutta method, algebraic stability, L-stability, discrete-gradient method, averaged vector field collocation.