A second-order L2-1s Crank-Nicolson difference method for two-dimensional time-fractional wave equations with variable coefficients

作者：Wang, Yuan-Ming^*; Zheng, Zi-Yun

来源：Computers and Mathematics with Applications, 2022, 118: 183-207.

DOI：10.1016/j.camwa.2022.05.018

• Fractional wave equation
• Polynomial interpolation approximation
• Crank-Nicolson technique
• Difference method
• Stability and convergence

Based on quadratic and linear polynomial interpolation approximations and the Crank-Nicolson technique, a new second-order difference approximation method of the Caputo fractional derivative of order alpha is an element of(1,2) is proposed. This method is different from the known second-order methods, which is called the L2-1(sigma) Crank-Nicolson method in this paper. Using the L2-1(sigma) Crank-Nicolson method of the Caputo fractional derivative, a fully discrete L2-1(sigma) Crank-Nicolson difference method for two-dimensional time-fractional wave equations with variable coefficients is developed. The unconditional stability and convergence of the method are rigorously proved. The optimal error estimates in the discrete L-2-norm and H-1-norm are obtained under a relatively weak regularity condition. The error estimates show that the method has the second-order convergence in both time and space for all alpha is an element of(1,2). In order to overcome the loss of the temporal convergence order caused by the stronger singularity of the exact solution at the initial time, a nonuniform L2-1(sigma) Crank-Nicolson difference method is also developed on a class of general nonuniform time meshes. Numerical results confirm the theoretical convergence result. The effectiveness of the nonuniform method for non-smooth solutions with the stronger singularity at the initial time is tested on a class of initially graded time meshes.