Time two-grid algorithm based on finite difference method for two-dimensional nonlinear fractional evolution equations

Abstract

In this paper, a time two-grid finite difference (FD) algorithm is proposed for solving two-dimensional nonlinear fractional evolution equations. In this time two-grid FD algorithm, a nonlinear FD system is solved on the time coarse grid of size τC. And the Lagrange's linear interpolation formula is applied to provide some useful values for the time fine grid of size τF. Then, a linear system is solved on the time fine grid. In the temporal direction, a backward Euler method is employed for the time derivative and a first order convolution quadrature rule is applied to discretize the fractional integral term. And the second order central difference quotient is considered for the spatial approximation. By means of the discrete energy method, we obtain the unconditional discrete L2 stability and convergence of order O(τC2+τF+hx2+hy2), where hx and hy are the spatial step sizes in the x direction and the y direction, respectively. Numerical examples are presented to show the feasibility and efficiency of the time two-grid FD algorithm.