Multiscale Modeling &Simulation, Volume 18, Issue 2, Page 999-1024, January 2020.
We consider numerics/asymptotics for the rotating nonlinear Klein--Gordon (RKG) equation, an important PDE in relativistic quantum physics that can model a rotating galaxy in Minkowski metric and serves also as a model, e.g., for a “cosmic superfluid.” First, we formally show that in the nonrelativistic limit RKG converges to coupled rotating nonlinear Schrödinger equations (RNLS), which are used to describe the particle-antiparticle pair dynamics. Investigations of the vortex state of RNLS are carried out. Second, we propose three different numerical methods to solve RKG from relativistic regimes to nonrelativistic regimes in polar and Cartesian coordinates. In relativistic regimes, a semi-implicit finite difference Fourier spectral method is proposed in polar coordinates where both rotation terms are diagonalized simultaneously. In nonrelativistic regimes, to overcome the fast temporal oscillations, we adopt the rotating Lagrangian coordinates and introduce two efficient multiscale methods with uniform accuracy, i.e., the multirevolution composition method and the exponential integrator. Various numerical results confirm (uniform) accuracy of our methods. Simulations of vortices dynamics are presented.

中文翻译：

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关于旋转非线性Klein-Gordon方程：非相对论极限和数值方法

2020年1月，《多尺度建模与仿真》，第18卷，第2期，第999-1024页。
我们考虑旋转非线性Klein-Gordon（RKG）方程的数值/渐近性，这是相对论量子物理学中的重要PDE，它可以用Minkowski度量建模旋转星系，并且还可以用作模型，例如“宇宙超流体”。首先，我们正式表明，在非相对论极限中，RKG收敛于耦合旋转非线性Schrödinger方程（RNLS），该方程用于描述粒子-反粒子对动力学。对RNLS的涡旋状态进行了研究。其次，我们提出了三种不同的数值方法来求解RKG，从极坐标和笛卡尔坐标的相对论形式到非相对论形式。在相对论体系中，提出了在极坐标系中同时对角旋转两个旋转项的半隐式有限差分傅立叶谱方法。在非相对论体系中，为了克服快速的时间振荡，我们采用旋转的拉格朗日坐标，并引入了两种有效的，均一精度的有效多尺度方法，即多重旋转合成方法和指数积分器。各种数值结果证实了我们方法的准确性。提出了涡流动力学模拟。