Unconditionally energy-stable linear convex splitting algorithm for the L2 quasicrystals

Abstract

Quasicrystals have extensive applications in material sciences. In this article, we develop an unconditional energy-dissipation-preserving, temporally second-order accurate, and linear method for solving a L2-gradient flow-based quasicrystal model. A truncated technique is introduced to regularize the nonlinear terms so that their second derivative is upper bounded. Based on the idea of the convex splitting method, we split free energy into the difference between two convex functionals. The convexity provides an appropriate choice of linear stabilization parameter. A linear and temporally second-order accurate algorithm is constructed utilizing the implicit-explicit Runge–Kutta method. In each time step, we can analytically prove that the proposed scheme satisfies the energy dissipation law. The Fourier spectral method is used to perform spatial discretization. Through numerical experiments, the accuracy, energy stability, and capability of the proposed method for quasicrystals are verified.