A Second-Order Length-Preserving and Unconditionally Energy Stable Rotational Discrete Gradient Method for Oseen-Frank Gradient Flows (Communications in Computational Physics, 2024)
We present a second-order strictly length-preserving and unconditionallyenergy-stable rotational discrete gradient (Rdg) scheme for the numerical approximation of the Oseen-Frank gradient flows with anisotropic elastic energy functional. Twoessential ingredients of the Rdg method are reformulation of the length constrainedgradient 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 proposedOseen-Frank discrete gradient satisfies the energy difference relation, thus the resultant Rdg scheme is energy stable. Numerical experiments demonstrate the efficiencyand accuracy of the proposed Rdg method and its capability for providing reliablesimulation results with highly disparate elastic coefficients.
Keywords: Nematic liquid crystal; Oseen-Frank gradient flow; energy stability; length-preservation; rotational discrete gradient method.