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Due to the lack of corresponding analysis on appropriate mapping operatorbetween two grids, high-order two-grid difference algorithms are rarelystudied. In this paper, we firstly discuss the boundedness of a local bi-cubicLagrange interpolation operator. And then, taking the semilinear parabolicequation as an example, we first construct a variable-step high-order nonlineardifference algorithm using compact difference technique in space and thesecond-order backward differentiation formula (BDF2) with variable temporalstepsize in time. With the help of discrete orthogonal convolution (DOC)kernels and a cut-off numerical technique, the unique solvability andcorresponding error estimates of the high-order nonlinear difference scheme areestablished under assumptions that the temporal stepsize ratio satisfies rk <4.8645 and the maximum temporal stepsize satisfies tau = o(h^1/2 ). Then, anefficient two-grid high-order difference algorithm is developed by combining asmall-scale variable-step high-order nonlinear difference algorithm on thecoarse grid and a large-scale variable-step high-order linearized differencealgorithm on the fine grid, in which the constructed piecewise bi-cubicLagrange interpolation mapping operator is adopted to project the coarse-gridsolution to the fine grid. Under the same temporal stepsize ratio restrictionrk < 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 thetwo-grid difference scheme is established if the coarse-fine grid stepsizessatisfy H = O(h^4/7). Finally, several numerical experiments are carried out todemonstrate the effectiveness and efficiency of the proposed scheme.

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