Convergence Analysis of a Second Order Convex Splitting Scheme for the Modified Phase Field Crystal Equation

TL;DR

This paper provides a rigorous convergence analysis of a second-order, energy-stable convex splitting scheme for the Modified Phase Field Crystal equation, establishing second-order accuracy in both time and space.

Contribution

It introduces a higher order consistency analysis to prove second-order convergence for a novel scheme applied to a generalized damped wave equation.

Findings

Proves second-order convergence in discrete L^a0(0,T; H^3) norm.

Establishes unconditional energy stability of the scheme.

Addresses accuracy challenges due to hyperbolic nature of the equation.

Abstract

In this paper we provide a detailed convergence analysis for an unconditionally energy stable, second-order accurate convex splitting scheme for the Modified Phase Field Crystal equation, a generalized damped wave equation for which the usual Phase Field Crystal equation is a special degenerate case. The fully discrete, fully second-order finite difference scheme in question was derived in a recent work [2]. An introduction of a new variable \psi, corresponding to the temporal derivative of the phase variable \phi, could bring an accuracy reduction in the formal consistency estimate, because of the hyperbolic nature of the equation. A higher order consistency analysis by an asymptotic expansion is performed to overcome this difficulty. In turn, second order convergence in both time and space is established in a discrete L^\infty (0,T; H^3) norm.