Unconditionally energy stable time marching numerical schemes for the modified phase field crystal equation

The modified phase field crystal (MPFC) equation is a sixth-order nonlinear damped wave equation modeling a viscoelastic response to perturbations to the density field. In this study, we propose first- and second-order time marching numerical schemes to solve the MPFC equation under the periodic boundary conditions. Recently, the scalar auxiliary variable (SAV) method has received considerable attention in solving of the gradient flow equations due to its advantage of preserving the energy dissipation law in a discrete sense. The proposed numerical schemes in this work are based on the exponential SAV (ESAV) method. More specifically, the scalar variable in the ESAV method is an exponential form rather than the square root function typically employed in the SAV approach. By making this distinctive change, the ESAV schemes are able to solve only one linear system with constant coefficients at each time step. In this paper, we first propose the first-order ESAV scheme based on the backward Euler method, then we develop the second-order ESAV scheme by utilizing the leapfrog approach. Theoretically, we present the unique solvability, mass conservation and unconditional energy stability of our proposed schemes. Moreover, a rigorous error analysis is provided for the second-order ESAV scheme. A comprehensive set of numerical simulations and examples are provided to validate the effectiveness of these schemes.