Convergence Analysis for Second Order Accurate Convex Splitting Schemes for the Periodic Nonlocal Allen-Cahn and Cahn-Hilliard Equations

TL;DR

This paper provides a rigorous convergence analysis of second order accurate, energy-stable numerical schemes for nonlocal Allen-Cahn and Cahn-Hilliard equations, establishing their stability and convergence properties.

Contribution

It introduces a detailed convergence proof for fully discrete second order schemes for nonlocal phase field models, including novel error estimates for the nonlinear terms.

Findings

Proves unconditional energy stability and unique solvability.

Establishes $O(s^3 + h^4)$ convergence in discrete norms.

Demonstrates convergence in maximum norm under standard constraints.

Abstract

In this paper we provide a detailed convergence analysis for fully discrete second order (in both time and space) numerical schemes for nonlocal Allen-Cahn (nAC) and nonlocal Cahn-Hilliard (nCH) equations. The unconditional unique solvability and energy stability ensures $\ell^4$ stability. The convergence analysis for the nAC equation follows the standard procedure of consistency and stability estimate for the numerical error function. For the nCH equation, due to the complicated form of the nonlinear term, a careful expansion of its discrete gradient is undertaken and an $H^{-1}$ inner product estimate of this nonlinear numerical error is derived to establish convergence. In addition, an a-priori $W^{1,\infty}$ bound of the numerical solution at the discrete level is needed in the error estimate. Such a bound can be obtained by performing a higher order consistency analysis by using asymptotic expansions for the numerical solution. Following the technique originally proposed by Strang (e.g., 1964), instead of the standard comparison between the exact and numerical solutions, an error estimate between the numerical solution and the constructed approximate solution yields an $O( s^3 + h^4)$ convergence in $\ell^\infty (0, T; \ell^2)$ norm, which leads to the necessary bound under a standard constraint $s \le C h$. Here, we also prove convergence of the scheme in the maximum norm under the same constraint.