Linear multi-step methods and their numerical stability for solving gradient flow equations

In this paper, linear multi-step methods are used to numerically solve gradient flow models, and the relations between different numerical stabilities (e.g., unconditional energy stability, A-stability and G-stability) of linear multi-step methods are discussed. First, we introduce the definition of the absolutely unconditional energy stability (AUES), in which the meaning of absoluteness is borrowed from the absolute stability of numerical methods, and we rigorously prove that the AUES is equivalent to the A-/G-stability of the scheme. Then, we obtain a new gradient flow system by using scalar auxiliary variable (SAV) approach with Lagrange multiplier. The linear multi-step method and the Fourier pseudo-spectral method are respectively used to discretize the temporal and spatial variables of the new gradient flow system, and its AUES is guaranteed by simple checking its A-/G-stability of the numerical scheme. Especially, we show that these schemes not only include the commonly-used backward Euler, Crank-Nicolson, second-order backward differentiation formula (BDF2) schemes, but also include some generalized Adams and Nystrom schemes. Finally, we apply these numerical schemes for solving gradient flow models, and ample numerical results are provided to demonstrate the high performance of the proposed schemes.

Automated summary

In this study, linear multi-step methods are applied to numerically solve gradient flow models, and the relationships between different numerical stabilities (e.g., unconditional energy stability, A-stability and G-stability) of linear multi-step methods are discussed. The concept of absolutely unconditional energy stability (AUES) is introduced, and its equivalence to the A-/G-stability of the scheme is rigorously proved. A new gradient flow system is obtained using the scalar auxiliary variable (SAV) approach with Lagrange multiplier. The AUES of the system is guaranteed by checking the A-/G-stability of the numerical scheme.