Energy-diminishing integration of gradient systems
Ernst Hairer and Christian Lubich
Abstract.
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.
Key Words.
Gradient flow, energy dissipation,
implicit Runge--Kutta method, algebraic stability, L-stability,
discrete-gradient method, averaged vector field collocation.