A New L2-Gradient Flow-Based Fractional-in-Space Modified Phase-Field Crystal Equation and Its Mass Conservative and Energy Stable Method

In this paper, we introduce a new fractional-in-space modified phase-field crystal equation based on the L2-gradient flow approach, where the mass of atoms is conserved by using a nonlocal Lagrange multiplier. To solve the L2-gradient flow-based fractional-in-space modified phase-field crystal equation, we present a mass conservative and energy stable method based on the convex splitting idea. Numerical examples together with standard tests in the classical H−1-gradient flow-based modified phase-field crystal equation are provided to illustrate the applicability of the proposed framework.