On Time-Splitting Pseudospectral Discretization for Nonlinear Klein-Gordon Equation in Nonrelativistic Limit Regime (Communications in Computational Physics, 2014)
In this work, we are concerned with a time-splitting Fourier pseudospectral(TSFP) discretization for the Klein-Gordon (KG) equation, involving a dimensionlessparameter ε∈(0,1]. In the nonrelativistic limit regime, the small ε produces high oscillationsin exact solutions with wavelength of O(ε2) in time. The key idea behind theTSFP is to apply a time-splitting integrator to an equivalent first-order system in time,with both the nonlinear and linear subproblems exactly integrable in time and, respectively,Fourier frequency spaces. The method is fully explicit and time reversible.Moreover, we establish rigorously the optimal error bounds of a second-order TSFPfor fixed ε = O(1), thanks to an observation that the scheme coincides with a type oftrigonometric integrator. As the second task, numerical studies are carried out, withspecial efforts made to applying the TSFP in the nonrelativistic limit regime, which aregeared towards understanding its temporal resolution capacity and meshing strategyfor O(ε2)-oscillatory solutions when 0 < ε ≪ 1. It suggests that the method has uniformspectral accuracy in space, and an asymptotic O(ε−2∆t2) temporal discretizationerror bound (∆t refers to time step). On the other hand, the temporal error bounds formost trigonometric integrators, such as the well-established Gautschi-type integratorin [6], are O(ε−4∆t2). Thus, our method offers much better approximations than theGautschi-type integrator in the highly oscillatory regime. These results, either rigorousor numerical, are valid for a splitting scheme applied to the classical relativistic NLSreformulation as well.