An efficient two-grid high-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation

Due to the lack of  corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this paper, we firstly discuss the boundedness of a local bi-cubic Lagrange interpolation operator. And then, taking the semilinear parabolic equation as an example, we first construct a  variable-step high-order nonlinear difference algorithm using compact difference technique in space and  the second-order backward differentiation formula with variable temporal stepsize in time. With the help of discrete orthogonal convolution kernels, temporal-spatial error splitting idea  and a cut-off numerical technique, the unique solvability, maximum-norm stability and corresponding  error estimate of the high-order nonlinear difference scheme are established under assumption that the temporal stepsize ratio satisfies $ r_{k} := \tau_{k}/\tau_{k-1} < 4.8645 $. Then, an efficient two-grid high-order difference algorithm is developed by combining a small-scale variable-step high-order nonlinear difference algorithm on the coarse grid and a large-scale variable-step high-order linearized difference algorithm on the fine grid, in which the constructed piecewise bi-cubic Lagrange interpolation mapping operator is adopted to project the coarse-grid solution to the fine grid. Under the same temporal stepsize ratio restriction $ r_{k} < 4.8645 $  on the variable temporal stepsize,  unconditional and optimal fourth-order in space and second-order in time maximum-norm error estimates of the two-grid difference scheme is established. Finally, several numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed scheme.