非线性Klein-Gordon方程的PML拟谱方法
Pseudospectral methods with PML for nonlinear Klein-Gordon equations in classical and non-relativistic regimes
摘要：
针对标准和非相对论性非线性Klein-Gordon方程（NKGE），推导了两种不同的具有有效伪谱数值格式的完全匹配层（PML）格式。提出并分析了一阶拟谱显式指数积分格式和二阶线性隐式预处理有限差分格式。为了获得高的空间精度，新的正则化Bermúdez型吸收轮廓的PML。结果表明，这两种格式都是有效的，但在与正则化Bermúdez型函数相结合的准谱方法框架下，为了保证精度，应采用线性隐格式.此外，在非相对论区域，数值算例表明正则化Bermúdez型吸收函数的误差对NKGE中的小参数$\varepsilon$不敏感.最后，通过一个二维的例子表明，该策略扩展到旋转NKGE的涡动力学是非常好的再现。
英文摘要：
Two different Perfectly Matched Layer (PML) formulations with efficient pseudo-spectral numerical schemes are derived for the standard and non-relativistic nonlinear Klein-Gordon equations (NKGE). A pseudo-spectral explicit exponential integrator scheme for a first-order formulation and a linearly implicit preconditioned finite-difference scheme for a second-order formulation are proposed and analyzed. To obtain a high spatial accuracy, new regularized Bermúdez type absorption profiles are introduced for the PML. It is shown that the two schemes are efficient, but the linearly implicit scheme should be preferred for accuracy purpose when used within the framework of pseudo-spectral methods combined with the regularized Bermúdez type functions. In addition, in the non-relativistic regime, numerical examples lead to the conclusion that the error related to regularized Bermúdez type absorption functions is insensitive to the small parameter $\varepsilon$ involved in the NKGE. The paper ends by a two-dimensional example showing that the strategy extends to the rotating NKGE where the vortex dynamics is very well-reproduced.
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