Arbitrary-order functionally fitted energy-diminishing methods for gradient systems
This work provides a new class of numerical integrators for gradient systems, which is important for simulations in physics, chemistry, and machine learning where energy dissipation is critical.
The authors developed novel numerical methods for gradient systems that are unconditionally energy-diminishing and can achieve arbitrarily high order, outperforming three existing methods in a numerical test.
It is well known that for gradient systems in Euclidean space or on a Riemannian manifold, the energy decreases monotonically along solutions. In this letter we derive and analyse functionally fitted energy-diminishing methods to preserve this key property of gradient systems. It is proved that the novel methods are unconditionally energy-diminishing and can achieve damping for very stiff gradient systems. We also show that the methods can be of arbitrarily high order and discuss their implementations. A numerical test is reported to illustrate the efficiency of the new methods in comparison with three existing numerical methods in the literature.