Effective numerical simulation of the Klein–Gordon–Zakharov system in the Zakharov limit

Abstract

Solving the Klein-Gordon-Zakharov (KGZ) system in the high-plasma frequency regime $c\gg1$ is numerically severely challenging due to the highly oscillatory nature or the problem. To allow reliable approximations classical numerical schemes require severe step size restrictions depending on the small parameter $c^{−2}$ . This leads to large errors and huge computational costs. In the singular limit $c\to\infty$ the Zakharov system appears as the regular limit system for the KGZ system. It is the purpose of this paper to use this approximation in the construction of an effective numerical scheme for the KGZ system posed on the torus in the highly oscillatory regime $c\gg1$. The idea is to filter out the highly oscillatory phases explicitly in the solution. This allows us to play back the numerical task to solving the non-oscillatory Zakharov limit system. The latter can be solved very efficiently without any step size restrictions. The numerical approximation error is then estimated by showing that solutions of the KGZ system in this singular limit can be approximated via the solutions of the Zakharov system and by proving error estimates for the numerical approximation of the Zakharov system. We close the paper with numerical experiments which show that this method is more effective than other methods in the high-plasma frequency regime $c\gg1$.