  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 (BDF2) with variable temporal
stepsize in time. With the help of discrete orthogonal convolution (DOC)
kernels and a cut-off numerical technique, the unique solvability and
corresponding error estimates of the high-order nonlinear difference scheme are
established under assumptions that the temporal stepsize ratio satisfies rk <
4.8645 and the maximum temporal stepsize satisfies tau = o(h^1/2 ). 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
rk < 4.8645 and a weaker maximum temporal stepsize condition tau = o(H^1.2 ),
optimal fourth-order in space and second-order in time error estimates of the
two-grid difference scheme is established if the coarse-fine grid stepsizes
satisfy H = O(h^4/7). Finally, several numerical experiments are carried out to
demonstrate the effectiveness and efficiency of the proposed scheme.
