针对半线性抛物方程的高效双网格高阶紧凑差分方案与变步长 BDF2 方法

Bingyin Zhang, Hongfei Fu
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引用次数: 0

摘要

由于缺乏对两网格间适当映射算子的相应分析,高阶两网格差分算法很少被研究。本文首先讨论了局部双立方拉格朗日插值算子的有界性。然后,以半线性抛物线方程为例,首先利用空间上的紧凑差分技术和时间上可变时间步长的二阶反向微分公式,构建了一种可变步长的高阶非线性差分算法。借助离散正交卷积核、时空误差分割思想和截断数值技术,在时间步长比满足 $ r_{k} := \tau_{k}/\tau_{k-1} < 4.8645 $ 的假设条件下,建立了高阶非线性差分方案的唯一可解性、最大正态稳定性和相应的误差估计。然后,结合粗网格上的小尺度变步长高阶非线性差分算法和细网格上的大尺度变步长高阶线性化差分算法,建立了一种高效的双网格高阶差分算法,其中采用了构建的片断双立方拉格朗日插值映射算子将粗网格解投影到细网格。在相同的时间步长比限制下,$ r_{k}< 4.8645 $ 的限制下,建立了双网格差分方案的无条件最优空间四阶和时间二阶最大正则误差估计。最后,通过几个数值实验证明了所提方案的有效性和效率。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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.
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