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In this paper, we present a novel second-order generalised rotationaldiscrete gradient scheme for numerically approximating the orthonormal framegradient flow of biaxial nematic liquid crystals. This scheme relies onreformulating the original gradient flow system into an equivalent generalised"rotational" form. A second-order discrete gradient approximation of the energyvariation is then devised such that it satisfies an energy difference relation.The proposed numerical scheme has two remarkable properties: (i) it strictlyobeys the orthonormal property of the tensor field and (ii) it satisfies theenergy dissipation law at the discrete level, regardless of the time stepsizes. We provide ample numerical results to validate the accuracy, efficiency,unconditional stability and SO(3)-preserving property of this scheme. Inaddition, comparisons of the simulation results between the biaxial orthonormalframe gradient flow model and uniaxial Oseen-Frank gradient flow are made todemonstrate the ability of the former to characterize non-axisymmetric localanisotropy.

A second-order SO(3)-preserving and energy-stable scheme for orthonormal frame gradient flow model of biaxial nematic liquid crystals