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.

翻译：由于缺乏对两网格间合适映射算子的相应分析，高阶双网格差分算法鲜有研究。本文首先讨论局部双三次拉格朗日插值算子的有界性。随后以半线性抛物方程为例，采用空间紧致差分技术和时间变步长二阶向后差分公式（BDF2），构建了变步长高阶非线性差分算法。借助离散正交卷积核与截断数值技术，在时间步长比满足rk<4.8645且最大时间步长满足τ=o(h^1/2)的假设下，建立了该高阶非线性差分格式的惟一可解性及相应误差估计。进而，通过将粗网格上的小规模变步长高阶非线性差分算法与细网格上的大规模变步长高阶线性化差分算法相结合，并采用所构造的分片双三次拉格朗日插值映射算子将粗网格解投影至细网格，发展了高效的双网格高阶差分算法。在相同时间步长比限制rk<4.8645及更宽松的最大时间步长条件τ=o(H^1.2)下，当粗-细网格步长满足H=O(h^4/7)时，建立了该双网格差分格式的空间四阶精度与时间二阶精度的最优误差估计。最后通过多个数值实验验证了所提格式的有效性与高效性。