A linear adaptive second-order backward differentiation formulation scheme for the phase field crystal equation

TL;DR

This paper introduces a linear, energy-stable, second-order adaptive BDF2 scheme for the phase field crystal equation, combining spectral spatial discretization with a novel auxiliary variable approach, validated through rigorous analysis and numerical tests.

Contribution

It develops a new linear adaptive BDF2 scheme with proven energy stability and second-order accuracy for the phase field crystal equation, using a first-order auxiliary variable approximation.

Findings

Unconditional energy stability under mild step size ratio constraints

Second-order accuracy on nonuniform meshes

Numerical validation confirms theoretical results

Abstract

In this paper, we present and analyze a linear fully discrete second order scheme with variable time steps for the phase field crystal equation. More precisely, we construct a linear adaptive time stepping scheme based on the second order backward differentiation formulation (BDF2) and use the Fourier spectral method for the spatial discretization. The scalar auxiliary variable approach is employed to deal with the nonlinear term, in which we only adopt a first order method to approximate the auxiliary variable. This treatment is extremely important in the derivation of the unconditional energy stability of the proposed adaptive BDF2 scheme. However, we find for the first time that this strategy will not affect the second order accuracy of the unknown phase function $\phi^{n}$ by setting the positive constant $C_{0}$ large enough such that $C_{0}\geq 1/\Dt.$ The energy stability of the adaptive BDF2 scheme is established with a mild constraint on the adjacent time step radio $\gamma_{n+1}:=\Dt_{n+1}/\Dt_{n}\leq 4.8645$. Furthermore, a rigorous error estimate of the second order accuracy of $\phi^{n}$ is derived for the proposed scheme on the nonuniform mesh by using the uniform $H^{2}$ bound of the numerical solutions. Finally, some numerical experiments are carried out to validate the theoretical results and demonstrate the efficiency of the fully discrete adaptive BDF2 scheme.