On the maximum bound principle and energy dissipation of exponential time differencing methods for the chiral liquid crystal blue phases

The blue phases are fascinating and complex states of chiral liquid crystals which can be modeled by a comprehensive framework of the Landau-de theory, satisfying energy dissipation and maximum bound principle. In this paper, we develop and analyze first and second order exponential time differencing numerical schemes for the gradient flow of the chiral liquid crystal blue phases, which preserve the maximum bound principle and energy dissipation unconditionally at the semi-discrete level. The fully discrete schemes are obtained coupled with the Fourier spectral method in space. And we propose a novel matrix-form Helmholtz basis transformation method to diagonalize the combined operator of the Laplacian and the curl operator, which is a key step in the implementation of the proposed schemes. Then by constructing auxiliary functions, we drive the $L^\infty$ boundedness of the numerical solutions and obtain the energy dissipation and the error estimates in $L^2$ and $L^\infty$ norm. Various numerical experiments are presented to validate the theoretical results and demonstrate the effectiveness of the proposed methods in simulating the dynamics of blue phases in chiral liquid crystals.