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Analysis and comparison of numerical methods for the Klein–Gordon equation in the nonrelativistic limit regime

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We analyze rigourously error estimates and compare numerically temporal/spatial resolution of various numerical methods for solving the Klein–Gordon (KG) equation in the nonrelativistic limit regime, involving a small parameter \({0 < {\varepsilon}\ll 1}\) which is inversely proportional to the speed of light. In this regime, the solution is highly oscillating in time, i.e. there are propagating waves with wavelength of \({O({\varepsilon}^2)}\) and O(1) in time and space, respectively. 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 h and time step τ as well as the small parameter \({{\varepsilon}}\). Based on the error bounds, in order to compute ‘correct’ solutions when \({0 < {\varepsilon}\ll1}\), the four FDTD methods share the same \({{\varepsilon}}\)-scalability: \({\tau=O({\varepsilon}^3)}\). Then we propose new numerical methods by using either Fourier pseudospectral or finite difference approximation for spatial derivatives combined with the Gautschi-type exponential integrator for temporal derivatives to discretize the KG equation. The new methods are unconditionally stable and their \({{\varepsilon}}\) -scalability is improved to τ = O(1) and \({\tau=O({\varepsilon}^2)}\) for linear and nonlinear KG equations, respectively, when \({0 < {\varepsilon}\ll1}\). Numerical results are reported to support our error estimates.

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Bao, W., Dong, X. Analysis and comparison of numerical methods for the Klein–Gordon equation in the nonrelativistic limit regime. Numer. Math. 120, 189–229 (2012). https://doi.org/10.1007/s00211-011-0411-2

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