A second-order length-preserving and unconditionally energy stable rotational discrete gradient method for Oseen-Frank gradient flows

TL;DR

This paper introduces a second-order, length-preserving, and energy-stable numerical scheme for simulating Oseen-Frank gradient flows, utilizing a novel discrete gradient tailored for anisotropic elastic energy.

Contribution

It develops a new second-order rotational discrete gradient method with a novel discrete gradient for Oseen-Frank flows, ensuring length preservation and unconditional energy stability.

Findings

The method is efficient and accurate in simulations.

It reliably handles highly disparate elastic coefficients.

The scheme maintains length and energy stability during simulations.

Abstract

We present a second-order strictly length-preserving and unconditionally energy-stable rotational discrete gradient (Rdg) scheme for the numerical approximation of the Oseen-Frank gradient flows with anisotropic elastic energy functional. Two essential ingredients of the Rdg method are reformulation of the length constrained gradient flow into an unconstrained rotational form and discrete gradient discretization for the energy variation. Besides the well-known mean-value and Gonzalez discrete gradients, we propose a novel Oseen-Frank discrete gradient, specifically designed for the solution of Oseen-Frank gradient flow. We prove that the proposed Oseen-Frank discrete gradient satisfies the energy difference relation, thus the resultant Rdg scheme is energy stable. Numerical experiments demonstrate the efficiency and accuracy of the proposed Rdg method and its capability for providing reliable simulation results with highly disparate elastic coefficients.