We propose and analyze a multiscale time integrator Fourier pseudospectral (MTI-FP) method for solving the Klein-Gordon (KG) equation with a dimensionless parameter $0<\varepsilon\leq1$ which is inversely proportional to the speed of light. In the nonrelativistic limit regime, i.e. $0<\varepsilon\ll1$, the solution to the KG equation propagates waves with amplitude at $O(1)$ and wavelength at $O(\varepsilon^2)$ in time and $O(1)$ in space, which causes significantly numerical burdens due to the high oscillation in time. The MTI-FP method is designed by adapting a multiscale decomposition by frequency (MDF) to the solution at each time step and applying an exponential wave integrator to the nonlinear Schrödinger equation with wave operator under well-prepared initial data for $\varepsilon^2$-frequency and $O(1)$-amplitude waves and a KG-type equation with small initial data for the reminder waves in the MDF. We rigorously establish two independent error bounds in $H^2$-norm to the MTI-FP method at $O(h^{m_0}+τ^2+\varepsilon^2)$ and $O(h^{m_0}+τ^2/\varepsilon^2)$ with $h$ mesh size, $τ$ time step and $m_0\ge2$ an integer depending on the regularity of the solution, which immediately imply that the MTI-FP converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at $O(τ)$ for all $\varepsilon\in(0,1]$ and optimally with quadratic convergence rate at $O(τ^2)$ in the regimes when either $\varepsilon=O(1)$ or $0<\varepsilon\le τ$. Numerical results are reported to confirm the error bounds and demonstrate the efficiency and accuracy of the MTI-FP method for the KG equation, especially in the nonrelativistic limit regime.
Symmetric high order Gautschi-type exponential wave integrators pseudospectral method for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime
Beijing Computational Science Research Center（北京计算科学研究中心）；IRMAR, Université de Rennes 1（雷恩第一大学 IRMAR）
Uniform and optimal error estimates of an exponential wave integrator sine pseudospectral method for the nonlinear Schrodinger equation with wave operator
National University of Singapore（新加坡国立大学）；Beijing Computational Science Research Center（北京计算科学研究中心）；University of Maryland（马里兰大学）
A uniformly accurate (UA) multiscale time integrator pseudospectral method for the Dirac equation in the nonrelativistic limit regime
National University of Singapore（新加坡国立大学）；Beijing Computational Science Research Center（北京计算科学研究中心）；Purdue University（普渡大学）；Université de Lorraine（洛林大学）；Inria Nancy-Grand Est（南锡大东部国家研究学院）
A Uniformly Accurate Multiscale Time Integrator for the Klein-Gordon-Schrödinger Equations in the Nonrelativistic Regime via Simplified Transmission Conditions
School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；Department of Mathematics, National University of Singapore（新加坡国立大学数学系）
A uniformly accurate multiscale time integrator for the nonlinear Klein-Gordon equation in the nonrelativistic regime via simplified transmission conditions
A uniformly accurate (UA) multiscale time integrator Fourier pseoduspectral method for the Klein-Gordon-Schrodinger equations in the nonrelativistic limit regime
High-order uniformly accurate fully discrete schemes with long time near-conservations for two nonlinear coupled dispersive models in the nonrelativistic regime
Xi’an Jiaotong University（西安交通大学）；Hebei Normal University（河北师范大学）
A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation
National University of Singapore（新加坡国立大学）；NUS Graduate School for Integrative Sciences and Engineering (NGS), National University of Singapore（新加坡国立大学整合科学与工程研究生学院）
Numerical methods and comparison for the Dirac equation in the nonrelativistic limit regime
National University of Singapore（新加坡国立大学）；Beijing Computational Science Research Center（北京计算科学研究中心）；Purdue University（普渡大学）；Institut Elie Cartan de Lorraine, Université de Lorraine, Inria Nancy-Grand Est（洛林埃利嘉当研究所，洛林大学，南锡大东部Inria）
Numerical approximation to Benjamin type equations. Generation and stability of solitary waves
University of Athens（雅典大学）；Institute of Applied & Computational Mathematics, FO.R.T.H.（应用与计算数学研究所，FO.R.T.H.）；University of Valladolid（巴利亚多利德大学）；Victoria University of Wellington（惠灵顿维多利亚大学）
Neural Multiscale Decomposition for Solving The Nonlinear Klein-Gordon Equation with Time Oscillation
National Center for Applied Mathematics（应用数学国家研究中心）；Tianjin University（天津大学）；School of Mechanical and Aerospace Engineering, Jilin University（吉林大学机械与航空航天工程学院）；Jilin University（吉林大学）
