In this talk, I will review our recent works on numerical methods and analysis for solving the nonlinear Klein-Gordon (KG) equation in the nonrelativistic limit regime, involving a small dimensionless parameter which is inversely proportional to the speed of light. In this regime, the solution is highly oscillating in time and the energy becomes unbounded, which bring significant difficulty in analysis and heavy burden in numerical computation. We begin with four frequently used finite difference time domain (FDTD) methods and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size and time step as well as the small parameter. Then we consider a numerical method by using spectral method for spatial derivatives combined with an exponential wave integrator (EWI) in the Gautschi-type for temporal derivatives to discretize the KG equation. Rigorious error estimates show that the EWI spectral method show much better temporal resolution than the FDTD methods for the KG equation in the nonrelativistic limit regime. In order to design a multiscale method for the KG equation, we establish error estimates of FDTD and EWI spectral methods for the nonlinear Schrodinger equation perturbed with a wave operator. Based on a large-small amplitude wave decompostion to the solution of the KG equation, a multiscale method is presented for discretizing the nonlinear KG equation in the nonrelativistic limit. Rigorous error estimates show that this multiscale method converges uniformly in spatia l/temporal discretization with respect to the small parameter for the nonlinear KG equation in the nonrelativistic limite regime. Finally, applications to several high oscillatory dispersive partial differential equations will be discussed.

[1] W. Bao and X. Dong, Analysis and comparison of numerical methods for Klein-Gordon equation

in nonrelativistic limit regime, Numer. Math., 120 (2012),  pp. 189-229.

[2] W. Bao and Y. Cai, Uniform error estimates of finite difference methods for the nonlinear

Schroedinger equation with wave operator, SIAM J Numer. Anal., 50 (2012), pp. 492-521.

[3] W. Bao and Y. Cai, Uniform and optimal error estimates of an exponential wave

integrator sine pseudospectral method for the nonlinear Schroedinger equation with wave operator,

SIAM J. Numer. Anal., 52 (2014), pp. 1103-1127.

[4] W. Bao, X. Dong and X. Zhao, Uniformly accurate multiscale time integrators for

highly oscillatory second order differential equations, J. Math. Study, 47 (20140, pp. 111-150.

[5] W. Bao, Y. Cai and X. Zhao, A uniformly accurate multiscale time integrator pseudospectral

method for the Klein-Gordon equation in the nonrelativistic limit regime,

SIAM J. Numer. Anal., 52 (2014), pp. 2488-2511.