Solving nonlinear Klein-Gordon equation with non-smooth solution by a geometric low-regularity integrator
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912253863788544 |
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| author | Wang, Bin Miao, Zhen Jiang, Yaolin |
| author_facet | Wang, Bin Miao, Zhen Jiang, Yaolin |
| contents | In this paper, we formulate and analyse a geometric low-regularity integrator for solving the nonlinear Klein-Gordon equation in the $d$-dimensional space with $d=1,2,3$. The integrator is constructed based on the two-step trigonometric method and thus it has a simple form. Error estimates are rigorously presented to show that the integrator can achieve second-order time accuracy in the energy space under the regularity requirement in $H^{1+\frac{d}{4}}\times H^{\frac{d}{4}}$. Moreover, the time symmetry of the scheme ensures its good long-time energy, momentum and action conservations which are rigorously proved by the technique of modulated Fourier expansions. A numerical test is presented and the numerical results demonstrate the superiorities of the new integrator over some existing methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_14062 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Solving nonlinear Klein-Gordon equation with non-smooth solution by a geometric low-regularity integrator Wang, Bin Miao, Zhen Jiang, Yaolin Numerical Analysis 35L70, 65M12, 65M15, 65M70 In this paper, we formulate and analyse a geometric low-regularity integrator for solving the nonlinear Klein-Gordon equation in the $d$-dimensional space with $d=1,2,3$. The integrator is constructed based on the two-step trigonometric method and thus it has a simple form. Error estimates are rigorously presented to show that the integrator can achieve second-order time accuracy in the energy space under the regularity requirement in $H^{1+\frac{d}{4}}\times H^{\frac{d}{4}}$. Moreover, the time symmetry of the scheme ensures its good long-time energy, momentum and action conservations which are rigorously proved by the technique of modulated Fourier expansions. A numerical test is presented and the numerical results demonstrate the superiorities of the new integrator over some existing methods. |
| title | Solving nonlinear Klein-Gordon equation with non-smooth solution by a geometric low-regularity integrator |
| topic | Numerical Analysis 35L70, 65M12, 65M15, 65M70 |
| url | https://arxiv.org/abs/2312.14062 |