| Issue |
ESAIM: M2AN
Volume 58, Number 2, March-April 2024
|
|
|---|---|---|
| Page(s) | 421 - 455 | |
| DOI | https://doi.org/10.1051/m2an/2024008 | |
| Published online | 04 April 2024 | |
An efficient two-grid high-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation
1
School of Mathematical Sciences, Ocean University of China, Qingdao, Shandong 266100, P.R. China
2
Laboratory of Marine Mathematics, Ocean University of China, Qingdao, Shandong 266100, P.R. China
* Corresponding author: fhf@ouc.edu.cn
Received:
23
June
2023
Accepted:
28
January
2024
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 rk := τk/τk−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 rk < 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.
Mathematics Subject Classification: 65M06 / 65M12 / 65M15 / 65M50
Key words: High-order two-grid difference algorithm / variable-step BDF2 method / piecewise bi-cubic Lagrange interpolation / DOC kernels / unique solvability / maximum-norm error estimate
© The authors. Published by EDP Sciences, SMAI 2024
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