Uniform Error Bounds of a Conservative Compact Finite Difference Method for the Quantum Zakharov System in the Subsonic Limit Regime

Abstract

In this paper, we consider a uniformly accurate compact finite difference method to solve the quantum Zakharov system (QZS) with a dimensionless parameter $0 < ε ≤ 1,$ which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e., when $0 < ε ≪ 1,$ the solution of QZS propagates rapidly oscillatory initial layers in time, and this brings significant difficulties in devising numerical algorithm and establishing their error estimates, especially as $0 < ε ≪ 1.$ The solvability, the mass and energy conservation laws of the scheme are also discussed. Based on the cut-off technique and energy method, we rigorously analyze two independent error estimates for the well-prepared and ill-prepared initial data, respectively, which are uniform in both time and space for $ε ∈ (0, 1]$ and optimal at the fourth order in space. Numerical results are reported to verify the error behavior.