Fully discrete, decoupled, second-order and unconditionally energy-stable DG–SAV–ZEC scheme for smectic-A liquid crystal hydrodynamics

Designing a fully discrete algorithm for smectic-A hydrodynamics that is at once linear, second-order-in-time, block-decoupled, and unconditionally stable remains highly nontrivial, especially in the presence of anisotropic dissipative stress. In this work, we develop a discontinuous Galerkin SAV–ZEC framework for the smectic-A hydrodynamic model that addresses this difficulty without nonlinear iterations or monolithic coupled solves. On the temporal side, a scalar auxiliary variable reformulation, a zero-energy-contribution treatment of the energy-neutral couplings, and a rotational pressure-correction step are combined to yield a block-decoupled second-order-in-time algorithm. On the spatial side, a thermodynamically consistent discontinuous Galerkin discretization is constructed using reconstructed gradient operators with lifting of interelement jumps and a discrete variational treatment of the constitutive relations. The resulting scheme preserves the coercivity of the discrete dissipative-stress operator, is uniquely solvable at each time step, and, for arbitrary time-step sizes, satisfies a fully discrete modified-energy dissipation law. Numerical experiments verify the expected second-order temporal accuracy and energy dissipation, and demonstrate stable numerical behavior in simulations of lamellar relaxation, strong layer reorganization, and defect-mediated dynamics in both two and three dimensions.