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