High Order Schemes for Gradient Flows

Abstract

First, two new classes of energy stable, high order accurate Runge-Kutta schemes for gradient flows in a very general setting are presented: a class of fully implicit methods that are unconditionally energy stable and a class of semi-implicit methods that are conditionally energy stable. The new schemes are developed as high order analogs 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 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 methods give a painless way to extend the existing schemes to high order accurate in time schemes while maintaining their stability. Additionally, we extend the schemes to gradient flows with solution dependent inner product. Here, the stability and consistency conditions of the methods are given and proved, specific examples of the schemes are given for second and third order accuracy, and convergence tests are performed to demonstrate the accuracy of the methods.

Next, two algorithms for simulating mean curvature motion are considered. First is the threshold dynamics algorithm of Merriman, Bence, and Osher. The algorithm is only first order accurate in the two-phase setting and its accuracy degrades further to half order in the multi-phase setting, a shortcoming it has in common with other related, more recent algorithms. As a first, rigorous step in addressing this shortcoming, two different second order accurate versions of two-phase threshold dynamics are presented. Unlike in previous efforts in this direction, both algorithms come with careful consistency calculations. The first algorithm is consistent with its limit (motion by mean curvature) up to second order in any space dimension. The second achieves second order accuracy only in dimension two but comes with a rigorous stability guarantee (unconditional energy stability) in any dimension -- a first for high order schemes of its type.

Finally, a level set method for multiphase curvature motion known as Voronoi implicit interface method is considered. Here, careful numerical convergence studies, using parameterized curves to reach very high resolutions in two dimensions are given. These tests demonstrate that in the unequal, additive surface tension case, the Voronoi implicit interface method does not converge to the desired limit. Then a variant that maintains the spirit of the original algorithm is presented. It appears to fix the non-convergence and as a bonus, the new variant extends the Voronoi implicit interface method to unequal mobilities.