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arXiv · 2312.14062

Solving nonlinear Klein-Gordon equation with non-smooth solution by a geometric low-regularity integrator

Bin Wang·Zhen Miao·Yaolin Jiang
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Abstract

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.

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Bin Wang, Zhen Miao, Yaolin Jiang. 2023-12-21. Solving nonlinear Klein-Gordon equation with non-smooth solution by a geometric low-regularity integrator. https://arxiv.org/abs/2312.14062

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