Variational Extrapolation of Implicit Schemes for General Gradient Flows

TL;DR

This paper presents a novel high order accurate, unconditionally energy stable scheme for gradient flows, extending the minimizing movements approach through a variational extrapolation technique that enhances existing methods.

Contribution

It introduces a variational extrapolation method to develop high order schemes for gradient flows, maintaining unconditional stability and extending classical minimizing movement approaches.

Findings

The schemes are unconditionally energy stable.

They achieve high order accuracy in time.

The approach generalizes existing methods to higher order.

Abstract

We introduce a class of unconditionally energy stable, high order accurate schemes for gradient flows in a very general setting. The new schemes are a high order analogue of the minimizing movements approach for generating a time discrete approximation to a gradient flow by solving a sequence of optimization problems. In particular, each step entails minimizing the associated energy of the gradient flow plus a movement limiter term that is, in the classical context of steepest descent with respect to an inner product, simply quadratic. A variety of existing unconditionally stable numerical methods can be recognized as (typically just first order accurate in time) minimizing movement schemes for their associated evolution equations, already requiring the optimization of the energy plus a quadratic term at every time step. Therefore, our approach gives a painless way to extend these to high order accurate in time schemes while maintaining their unconditional stability. In this sense, it can be viewed as a variational analogue of Richardson extrapolation.