Efficient $BDF_k$$(k = 3, 4, 5)$ Schemes for the Cahn-Hilliard Model: Construction, Dissipation Preservation and Convergence Analysis

Source abstract

Abstract. It is widely acknowledged that analyzing concisely the convergence of high order backward differentiation formula (BDF) for nonlinear dissipative partial differential equations is an interesting and challenging issue. In this paper, we derive a type of efficient, high-order accurate and unconditionally energy stable numerical schemes for solving the Cahn-Hilliard model by combining the high order backward differentiation formula with the scalar auxiliary variable approach. A modified energy dissipation law at the discrete levels is proved by using the discrete gradient decomposition of the $k$-th order BDF formula $(k = 3, 4, 5)$. The $H^2$ bound of the numerical solution is obtained based on the convolution form of the numerical schemes. Furthermore, an optimal $L^2$ norm error estimate is established concisely by utilizing the new developed estimates that handle the nonlinear term and scalar auxiliary variable. Numerical examples are presented to verify the theoretical analysis and demonstrate the efficiency of the proposed numerical method.