Uniform error bounds of a conservative compact finite difference method for the quantum Zakharov system in the subsonic limit regime. (English) Zbl 07803031
Summary: In this paper, we consider a uniformly accurate compact finite difference method to solve the quantum Zakharov system (QZS) with a dimensionless parameter \(0 < \varepsilon \leq 1\), which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e., when \(0 < \varepsilon \ll 1\), 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 \(0 < \varepsilon \ll 1\). 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 \(\varepsilon \in (0, 1]\) and optimal at the fourth order in space. Numerical results are reported to verify the error behavior.
MSC:
| 65-XX | Numerical analysis |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
Keywords:
quantum Zakharov system; subsonic limit; compact finite difference method; uniformly accurate; error estimateReferences:
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