On nested Picard iterative integrators for highly oscillatory second-order differential equations
On nested Picard iterative integrators for highly oscillatory second-order differential equations
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DOI：
10.1007/s11075-022-01317-8
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发表时间：
2022-05
影响因子：
2.1
通讯作者：
Yan Wang
中科院分区：
文献类型：
--
作者：
Yan Wang
This paper is devoted to the construction and analysis of uniformly accurate (UA) nested Picard iterative integrators (NPI) for highly oscillatory second-order differential equations. The equations involve a dimensionless parameterε∈ (0,1], and their solutions are highly oscillatory in time with wavelength at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\boldsymbol {\mathcal {O}}(\varepsilon ^{2})$\end{document}, which brings severe burdens in numerical computation whenε≪ 1. In this work, we first propose two NPI schemes for solving a differential equation. The schemes are uniformly first- and second-order accurate for allε∈ (0,1]. Moreover, they are super convergent when the time-step size is smaller thanε2. Then, the schemes are generalized to a system of differential equations with the same uniform accuracies. Error bounds are rigorously established and numerical results are reported to confirm the error estimates.
This paper is devoted to the construction and analysis of uniformly accurate (UA) nested Picard iterative integrators (NPI) for highly oscillatory second-order differential equations. The equations involve a dimensionless parameterε∈ (0,1], and their solutions are highly oscillatory in time with wavelength at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\boldsymbol {\mathcal {O}}(\varepsilon ^{2})$\end{document}, which brings severe burdens in numerical computation whenε≪ 1. In this work, we first propose two NPI schemes for solving a differential equation. The schemes are uniformly first- and second-order accurate for allε∈ (0,1]. Moreover, they are super convergent when the time-step size is smaller thanε2. Then, the schemes are generalized to a system of differential equations with the same uniform accuracies. Error bounds are rigorously established and numerical results are reported to confirm the error estimates.