Analysis of two conservative fourth-order compact finite difference schemes for the Klein-Gordon-Zakharov system in the subsonic limit regime

Abstract

We propose two conservative fourth-order compact finite difference (CFD4C) schemes and give the rigorous error analysis for the Klein-Gordon-Zakharov system (KGZS) with ε∈(0,1] being a small parameter. In the case 0<ε≪1, i.e., the subsonic limit regime, the solution of KGZS propagates waves with wavelength O(ε) in time and O(1) in space, respectively, with amplitude at O(εα†) with α†=min⁡{α,β+1,2}, where α and β describe the incompatibility between the initial data of the KGZS and the limiting Klein-Gordon equation (KGE) as ε→0+ and satisfy α≥0,β+1≥0. Different from the standard KGZS, the oscillation in time becomes the main difficulty in constructing numerical schemes and giving the error analysis for KGZS in this regime. We prove that the schemes conserve the discrete energies. By defining a discrete ‘energy’, applying cut-off technique and considering the convergence rate of the KGZS to the limiting KGE, we obtain two independent error estimates with the bounds at O(h4/ε1−α⁎+τ2/ε3−α†) and O(h4+τ2+εα⁎) where α⁎=min⁡{1,α,1+β}, τ is time step and h is mesh size. Hence, we get uniformly accurate error estimates with the bounds at O(h4+τ2/(4−α†)) when α≥1 and β≥0, and O(h4α⁎+τ2α⁎/3) when 0<α<1 or −1<β<0. While for α=0 or β=−1, the meshing requirements of CFD4Cs are h=O(ε1/4) and τ=O(ε3/2) for 0<ε≪1. The compact schemes provide much better spatial resolution compared with the usual second order finite difference schemes. Our numerical results support energy-conserving properties and the results of the convergence analysis.