Abstract In this paper, we first present the nonlinear stability and convergence of extended Runge-Kutta-Nystrom (ERKN) integrators for nonlinear multi-frequency highly oscillatory second-order ODEs with a takanami number and dominant frequency matrix. We then rigorously analyse the global errors of the blend of the ERKN time integrators and the Fourier pseudospectral spatial discretisation (ERKN-FP) when applied to semi-linear wave equations. The theoretical results based on the energy techniques show that the nonlinear stability and the global error bounds are entirely independent of the takanami number, the dominant frequency matrix, and the spatial mesh size. The analysis also provides a new perspective on the ERKN time integrators. That is, the ERKN-FP methods are free from the restriction on the CFL condition when applied to semi-linear wave equations. Numerical experiments verify our theoretical analysis results and demonstrate the remarkable superiority of the ERKN time integrators in comparison with traditional time-integration methods in the literature.

中文翻译：

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用于求解非线性多频高振荡二阶常微分方程的 ERKN 积分器的非线性稳定性和收敛性，并应用于半线性波动方程

摘要 在本文中，我们首先介绍了具有高波数和主频率矩阵的非线性多频高振荡二阶常微分方程的扩展 Runge-Kutta-Nystrom (ERKN) 积分器的非线性稳定性和收敛性。然后，当应用于半线性波动方程时，我们严格分析了 ERKN 时间积分器和傅立叶伪谱空间离散化 (ERKN-FP) 混合的全局误差。基于能量技术的理论结果表明非线性稳定性和全局误差界限完全独立于高波数、主频率矩阵和空间网格大小。该分析还提供了有关 ERKN 时间积分器的新视角。那是，ERKN-FP 方法在应用于半线性波动方程时不受 CFL 条件的限制。数值实验验证了我们的理论分析结果，并证明了 ERKN 时间积分器与文献中的传统时间积分方法相比具有显着的优越性。