Energy stability and convergence of SAV block-centered finite difference method for gradient flows

Abstract

We present in this paper construction and analysis of a block-centered finite difference method for the spatial discretization of the scalar auxiliary variable Crank-Nicolson scheme (SAV/CN-BCFD) for gradient flows, and show rigorously that scheme is second-order in both time and space in various discrete norms. When equipped with an adaptive time strategy, the SAV/CN-BCFD scheme is accurate and extremely efficient. Numerical experiments on typical Allen-Cahn and Cahn-Hilliard equations are presented to verify our theoretical results and to show the robustness and accuracy of the SAV/CN-BCFD scheme.

Modeling and Simulation of a Ternary System for Macromolecular Microsphere Composite Hydrogels

“…Nevertheless, only the modified energy is unconditionally stable but not the original one. More applications can be found in [27,68] and the convergence and error analysis are presented in [31,34,44].…”

Unconditional Stability and Optimal Error Estimates of Euler Implicit/Explicit-SAV Scheme for the Navier–Stokes Equations

“…where Δt is the time step, C > 0 is a general constant, independents of Δt and mesh size h. The scalar auxiliary variable (SAV) method was developed by Shen and his co-authors [20,21,30,31] for the gradient flow. This method can be considered as an extension and improvement of the invariant energy quadratization (IEQ) method given in [36,37,39].…”

Efficient unconditionally stable numerical schemes for a modified phase field crystal model with a strong nonlinear vacancy potential

“…Yang et al developed invariant energy quadratization (IEQ) approach [56,59,60,[63][64][65], which leads to linear, unconditionally energy stable numerical schemes [58]. The scalar auxiliary variable (SAV) approach [44][45][46] was proposed by Shen and his collaborators, and has been applied to construct linearly energy stable numerical schemes [32,33,40]. In References [7,8,21,22,26,[36][37][38][39], the authors extended the IEQ or SAV approach to design linear, energy stable numerical schemes.…”

“…We also derive error estimates in time for the first-order numerical scheme. For the SAV approach, rigorous error estimates have been carried out [31][32][33]44]. For the IEQ approach, there have been convergence analysis for Cahn-Hilliard type equations [61,72] and the PFC model [36].…”