Uniform Error Bounds of a Conservative Compact Finite Difference Method for the Quantum Zakharov System in the Subsonic Limit Regime (Journal of Computational Mathematics, 2023)
In this paper, we consider a uniformly accurate compact finite difference method tosolve 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 establishingtheir error estimates, especially as $0 < ε ≪ 1.$ The solvability, the mass and energyconservation laws of the scheme are also discussed. Based on the cut-off technique andenergy method, we rigorously analyze two independent error estimates for the well-preparedand 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 verifythe error behavior.
Keywords: Quantum Zakharov system; Subsonic limit; Compact finite difference method; Uniformly accurate; Error estimate.