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

TL;DR

This paper introduces high-order, energy-diminishing numerical methods for gradient systems that preserve the natural energy decay property, demonstrating their effectiveness through theoretical analysis and numerical tests.

Contribution

It develops and analyzes arbitrarily high-order, energy-diminishing methods specifically designed for gradient systems, ensuring energy decay and damping for stiff problems.

Findings

Methods are unconditionally energy-diminishing

Achieve damping for very stiff gradient systems

Numerical tests show improved efficiency over existing methods

Abstract

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.