We study numerical methods for the rotating nonlinear Klein-Gordon (RKG) equation, a fundamental model in relativistic quantum physics, which exhibits highly oscillatory multiscale behavior due to the presence of a small parameter {\epsilon}. The RKG equation models rotating galaxies under the Minkowski metric and also provides an effective description of phenomena such as cosmic superfluids. This work focuses on the development and rigorous analysis of structure-preserving Galerkin finite element methods (FEMs) for the RKG equation. A central challenge is that the rotational terms prevent traditional nonconforming FEMs from simultaneously conserving energy and charge. By employing a conservation-adjusting technique, we construct a consistent structure-preserving algorithm applicable to both conforming and nonconforming FEMs. Moreover, we provide a comprehensive convergence analysis, establishing unconditional optimal and high-order accuracy error estimates. These theoretical results are further validated through extensive numerical experiments, which demonstrate the accuracy, efficiency, and robustness of the structure-preserving schemes. Finally, simulations of vortex dynamics, ranging from the relativistic to the nonrelativistic regimes, are presented to illustrate vortex creation, relativistic effects on bound states, and interactions of vortex pairs.

翻译：本文研究了旋转非线性 Klein-Gordon (RKG) 方程的数值方法。该方程是相对论量子物理学中的一个基本模型，由于小参数 {\epsilon} 的存在，表现出高度振荡的多尺度行为。RKG 方程在闵可夫斯基度规下模拟旋转星系，同时也为宇宙超流体等现象提供了有效描述。本工作的重点是为 RKG 方程开发和严格分析保结构的 Galerkin 有限元方法 (FEMs)。一个核心挑战在于，旋转项的存在使得传统的非协调有限元法无法同时守恒能量和电荷。通过采用守恒调整技术，我们构建了一种适用于协调与非协调有限元法的一致保结构算法。此外，我们提供了全面的收敛性分析，建立了无条件最优及高阶精度的误差估计。这些理论结果通过大量数值实验得到了进一步验证，实验证明了保结构格式的精确性、高效性和鲁棒性。最后，我们展示了从相对论到非相对论区域的涡旋动力学模拟，用以说明涡旋的产生、相对论效应对束缚态的影响以及涡旋对的相互作用。