A direct PinT algorithm for higher-order nonlinear time-evolution equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911205338120192 |
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| author | Zhong, Shun-Zhi Zhao, Yong-Liang Shu, Qian-Yu |
| author_facet | Zhong, Shun-Zhi Zhao, Yong-Liang Shu, Qian-Yu |
| contents | Higher-order nonlinear time-evolution equations have widespread applications in science and engineering, such as in solid mechanics, materials science, and fluid mechanics. This paper mainly studies a direct time-parallel algorithm for solving time-dependent differential equations of orders 1 to 3. Different from the traditional time-stepping approach, we directly solve the all-at-once system from higher-order evolution equations by diagonalization the time discretization matrix $B$. Based on the connection between the characteristic equation and Chebyshev polynomials, we give explicit formulas for the eigenvector matrix $V$ of $B$ and its inverse $V^{-1}$. We prove that $Cond_2\left( V \right) =\mathcal{O} \left( n^3 \right)$, where $n$ is the number of time steps. A direct parallel-in-time algorithm is designed by exploring the structure of the spectral decomposition of $B$. Numerical experiments are provided to show the significant computational speedup of the proposed algorithm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_05743 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A direct PinT algorithm for higher-order nonlinear time-evolution equations Zhong, Shun-Zhi Zhao, Yong-Liang Shu, Qian-Yu Numerical Analysis Higher-order nonlinear time-evolution equations have widespread applications in science and engineering, such as in solid mechanics, materials science, and fluid mechanics. This paper mainly studies a direct time-parallel algorithm for solving time-dependent differential equations of orders 1 to 3. Different from the traditional time-stepping approach, we directly solve the all-at-once system from higher-order evolution equations by diagonalization the time discretization matrix $B$. Based on the connection between the characteristic equation and Chebyshev polynomials, we give explicit formulas for the eigenvector matrix $V$ of $B$ and its inverse $V^{-1}$. We prove that $Cond_2\left( V \right) =\mathcal{O} \left( n^3 \right)$, where $n$ is the number of time steps. A direct parallel-in-time algorithm is designed by exploring the structure of the spectral decomposition of $B$. Numerical experiments are provided to show the significant computational speedup of the proposed algorithm. |
| title | A direct PinT algorithm for higher-order nonlinear time-evolution equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2507.05743 |