Improved uniform error bounds of a Lawson-type exponential wave integrator method for the Klein-Gordon-Dirac equation

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For the Klein-Gordon-Dirac equation (KGDE) with small coupling constant ε∈(0,1], we propose a Lawson-type exponential wave integrator Fourier pseudo-spectral (LEWIFP) method and establish the improved uniform error bounds in the time domain at O(1/ε). We first convert the KGDE to a coupled system and then consider LEWIFP method for the coupled system. The LEWIFP method is proved to be time symmetric which is an important structure in numerical geometric integration. Through careful and rigorous convergence analysis, we establish the error bounds O(hm+ετ2+τ0m) for the full-discretization, where m is determined by the regularity conditions. If the solution is sufficiently smooth with m sufficiently large, we obtain the errors with improved uniform bounds at O(hm+ετ2) in the long-time domain up to O(1/ε). These error bounds are much better than the classical bounds O(hm+τ2) provided by the traditional analysis for the non-Lawson-type exponential wave integrators equipped with Fourier pseudo-spectral method. Combined with the classical analysis tools such as mathematical induction and energy method, we complete our error analysis by adopting the regularity compensation oscillation (RCO) technique which controls the high frequency modes by the regularity of the solution and low frequency modes by phase cancellation. By applying the LEWIFP method to some problems, we show the numerical results to support our error bounds. In addition, the numerical results also show that the discrete mass and energy are stable in the time domain which is long enough. Finally we extend our method to the oscillatory problem.