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In this paper, we are concerned with symmetric integrators for the nonlinearrelativistic Klein--Gordon (NRKG) equation with a dimensionless parameter$0<\varepsilon\ll 1$, which is inversely proportional to the speed of light.The highly oscillatory property in time of this model corresponds to theparameter $\varepsilon$ and the equation has strong nonlinearity when $\eps$ issmall. There two aspects bring significantly numerical burdens in designingnumerical methods. We propose and analyze a novel class of symmetricintegrators which is based on some formulation approaches to the problem,Fourier pseudo-spectral method and exponential integrators. Two practicalintegrators up to order four are constructed by using the proposed symmetricproperty and stiff order conditions of implicit exponential integrators. Theconvergence of the obtained integrators is rigorously studied, and it is shownthat the accuracy in time is improved to be $\mathcal{O}(\varepsilon^{3}\hh^2)$ and $\mathcal{O}(\varepsilon^{4} \hh^4)$ for the time stepsize $\hh$.The near energy conservation over long times is established for the multi-stageintegrators by using modulated Fourier expansions. These theoretical resultsare achievable even if large stepsizes are utilized in the schemes. Numericalresults on a NRKG equation show that the proposed integrators have improveduniform error bounds, excellent long time energy conservation and competitiveefficiency.

Symmetric integrators with improved uniform error bounds and long-time  conservations for the nonlinear Klein-Gordon equation in the nonrelativistic  limit regime