On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime

Abstract

This work concerns the time averaging techniques for the nonlinear Klein-Gordon (KG) equation in the nonrelativistic limit regime which have recently gained a lot of attention in numerical analysis. This is due to the fact that the solution becomes highly-oscillatory in time in this regime which causes the breakdown of classical integration schemes. To overcome this numerical burden various novel numerical methods with excellent efficiency were derived in recent years. The construction of each method thereby requests essentially the averaged model of the problem. However, the averaged model of each approach is found by different kinds of asymptotic approximation techniques reaching from the modulated Fourier expansion over the multiscale expansion by frequency up to the Chapman-Enskog expansion. In this work we give a first comparison of these recently introduced asymptotic series, reviewing their approximation validity to the KG in the asymptotic limit, their smoothness assumptions as well as their geometric properties, e.g., energy conservation and long-time behaviour of the remainder.2010 Mathematics Subject Classification. 35Q40, 34C29, 34E05, 81Q05. Key words and phrases. Nonlinear Klein-Gordon equation, nonrelativistic limit regime, time averaging, higher order expansion, long-time behavior. 2841 2842 KATHARINA SCHRATZ AND XIAOFEI ZHAO The so-called nonrelativistic limit ε → 0 of the KG equation (1.1) has been extensively studied in literature from a physical and mathematical point of view. Nowadays it is well understood that the KG equation converges to a nonlinear Schrödinger (NLS) equation when ε tends to zero. In Section 2 below we present the detailed structure of the NLS limit system. For recent analytic approximations results we refer to [36,33,34], and in the context of Birkhoff normal form transformations in particular to the recent work [38], as well as the references therein. From a numerical point of view, however, the Klein-Gordon equation in the nonrelativistic was an open problem for a long time. Classical methods are not able to resolve the highly-oscillatory nature of the solution which leads to severe step size restrictions (at least at order τ = O(ε −2 )) and huge computational costs ([6]). This failure of classical integration schemes triggered intensive studies on the numerical averaging of the model, serving to describe better asymptotic behaviour of the solution and to design efficient numerical approximations. Based on this idea, very recently novel numerical integrators were proposed which allow us to numerically solve the KG equation (1.1) in the non relativistic regime ε → 0, capturing the highly-oscillatory behaviour of the solution.Three novel schemes for the Klein-Gordon equation in the non relativistic limit regime were presented so far in literature [28,5,18]. Each of the proposed numerical schemes for KG is essentially based on an asymptotic expansion techinque for the averaged model, and each of the asymptotic expansion is mathematically obtained by different analytic techn...

Pseudospectral methods with PML for nonlinear Klein-Gordon equations in classical and non-relativistic regimes

“…It is well-known that as ε → 0, the solution of the NKGE (3.1) contains rapid oscillations in time [15,33,34,39]:…”

Derivative-free high-order uniformly accurate schemes for highly oscillatory systems

“…So far in the literature, UA methods have been constructed by different techniques by several authors [1,3,4,6,9,10,11,12,13,15,16,18]. It is out of the scope of this paper to describe all of them and for an extensive comparison, we refer the interested reader to [25]. However, all these methods have in common that they necessitate not only an analytic expression of f θ in (1.1), but also higher-order derivatives ∂ p u f θ in explicit form: instead of solving directly (1.1) the system is first transformed into a nonstiff one which is more amenable to numerical integration.…”