A third-order trigonometric integrator with low regularity for the semilinear Klein-Gordon equation
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| Format: | Preprint |
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2024
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| _version_ | 1866910711472455680 |
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| author | Wang, Bin Jiang, Yaolin |
| author_facet | Wang, Bin Jiang, Yaolin |
| contents | In this paper, we propose and analyse a novel third-order low-regularity trigonometric integrator for the semilinear Klein-Gordon equation with non-smooth solution in the $d$-dimensional space, where $d=1,2,3$. The integrator is constructed based on the full use of Duhamel's formula and the employment of a twisted function tailored for trigonometric integrals. Robust error analysis is conducted, demonstrating that the proposed scheme achieves third-order accuracy in the energy space under a weak regularity requirement in $H^{1+\max(μ,1)}(\mathbb{T}^d)\times H^{\max(μ,1)}(\mathbb{T}^d)$ with $μ> \frac{d}{2}$. A numerical experiment shows that the proposed third-order low-regularity integrator is much more accurate than some well-known exponential integrators of order three for approximating the Klein-Gordon equation with non-smooth solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19540 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A third-order trigonometric integrator with low regularity for the semilinear Klein-Gordon equation Wang, Bin Jiang, Yaolin Numerical Analysis 35L70, 65M12, 65M15, 65M70 In this paper, we propose and analyse a novel third-order low-regularity trigonometric integrator for the semilinear Klein-Gordon equation with non-smooth solution in the $d$-dimensional space, where $d=1,2,3$. The integrator is constructed based on the full use of Duhamel's formula and the employment of a twisted function tailored for trigonometric integrals. Robust error analysis is conducted, demonstrating that the proposed scheme achieves third-order accuracy in the energy space under a weak regularity requirement in $H^{1+\max(μ,1)}(\mathbb{T}^d)\times H^{\max(μ,1)}(\mathbb{T}^d)$ with $μ> \frac{d}{2}$. A numerical experiment shows that the proposed third-order low-regularity integrator is much more accurate than some well-known exponential integrators of order three for approximating the Klein-Gordon equation with non-smooth solutions. |
| title | A third-order trigonometric integrator with low regularity for the semilinear Klein-Gordon equation |
| topic | Numerical Analysis 35L70, 65M12, 65M15, 65M70 |
| url | https://arxiv.org/abs/2403.19540 |