A time two-grid algorithm for two-dimensional nonlinear time-fractional partial integro-differential equations

Abstract

In this paper, a temporal second order two-grid difference scheme is proposed for the two-dimensional nonlinear time-fractional partial integro-differential equations with a weakly singular kernel. The first-order backward difference and L1 formula are used in the temporal direction to estimate the first level of time, the L2−1σ formula and L1-type formula are used in the temporal direction for later time steps, and the central difference formula is used in the spatial directions. To improve the computational efficiency of nonlinear system, an efficient time two-grid algorithm is proposed. This algorithm firstly solves a nonlinear system on the coarse grid, and then the Lagrangian linear interpolation is applied on the coarse grid to estimate the function values on the fine grid. The stability and convergence of the two-grid difference scheme are analyzed by the energy method. The convergence order of the two-grid difference scheme is O(τF2+τC4+hx2+hy2), where τF and τC are the time step sizes of fine grid and coarse grid respectively, while hx and hy are the space step sizes. Numerical experiments show that the accuracy of the theoretical analysis and the efficiency of the two-grid algorithm.