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It is well known that for gradient systems in Euclidean space or on aRiemannian manifold, the energy decreases monotonically along solutions. Inthis letter we derive and analyse functionally fitted energy-diminishingmethods to preserve this key property of gradient systems. It is proved thatthe novel methods are unconditionally energy-diminishing and can achievedamping for very stiff gradient systems. We also show that the methods can beof arbitrarily high order and discuss their implementations. A numerical testis reported to illustrate the efficiency of the new methods in comparison withthree existing numerical methods in the literature.

Arbitrary-order functionally fitted energy-diminishing methods for gradient systems