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__Efficient time integration of the Klein-Gordon equation in the non-relativistic limit regime__

K. Schratz and E. Faou

**Abstract**

We consider the Klein-Gordon equation in the non-relativistic limit regime, i.e. the speed of light $c$ formally tending to infinity. Due to the highly-oscillatory nature of the solution in this regime, its numerical simulation is very delicate. Here we will construct an asymptotic expansion for the exact solution in terms of the small parameter $c^{-2}$. We will see that for sufficiently smooth initial values this asymptotic expansion exists up to an arbitrary order, and can be reconstructed by superposing highly-oscillatory terms to cascaded solutions of $c$-independent Schrödinger-like systems. The numerical advantage is that the high-oscillations in the exact solution can be filtered out explicitly and the numerical task reduces to solving the non-oscillatory Schrödinger-like limit systems, which can be carried out very efficiently without any additional time-step restriction.