A new space-fractional modified phase field crystal equation and its numerical algorithm

In this paper, we develop a new space-fractional modified phase field crystal equation which has some similar properties including the mass conservation and energy dissipation. Then, we propose a second-order scheme based on a new Lagrange multiplier method that conserves the mass and dissipates the energy. For the new method, there are only two decoupled linear equations with constant coefficients and one nonlinear algebraic system to be solved at each time step which makes it efficient. Finally, we give some numerical experiments to verify the accuracy and stability of the proposed methods.