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arXiv 2006.09370math.AP
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对数型Klein-Gordon方程的两种正则化保能量有限差分法

Two regularized energy-preserving finite difference methods for the logarithmic Klein-Gordon equation

Jingye Yan, Xu Qian, Hong Zhang, Songhe Song

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arXiv admin note: text overlap with arXiv:2006.08079
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摘要：

提出并分析了对数Klein-Gordon方程（LogKGE）的两种保能量正则化有限差分法。为了避免LogKGE的对数非线性引起的奇异性，我们提出了一种带有小调节参数$0<\varepsilon\ll1$的正则化对数Klein-Gordon方程（RLogKGE），以$O(\varepsilon)$的收敛阶逼近LogKGE.采用能量法、逆不等式和非线性截断技术对数值解进行了定界，给出了两种格式的误差界$O(h^{2}+\frac{τ^{2}}{\varepsilon^{2}})$与网格步长$h$、时间步长$τ$和参数$\varepsilon$的关系。数值结果支持我们的结论。

英文摘要：

We present and analyze two regularized finite difference methods which preserve energy of the logarithmic Klein-Gordon equation (LogKGE). In order to avoid singularity caused by the logarithmic nonlinearity of the LogKGE, we propose a regularized logarithmic Klein-Gordon equation (RLogKGE) with a small regulation parameter $0<\varepsilon\ll1$ to approximate the LogKGE with the convergence order $O(\varepsilon)$. By adopting the energy method, the inverse inequality, and the cut-off technique of the nonlinearity to bound the numerical solution, the error bound $O(h^{2}+\frac{τ^{2}}{\varepsilon^{2}})$ of the two schemes with the mesh size $h$, the time step $τ$ and the parameter $\varepsilon$. Numerical results are reported to support our conclusions.

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