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Volume 42, Issue 1
Uniform Error Bounds of a Conservative Compact Finite Difference Method for the Quantum Zakharov System in the Subsonic Limit Regime

Gengen Zhang & Chunmei Su

DOI: 10.4208/jcm.2204-m2022-0001

J. Comp. Math., 42 (2024), pp. 289-312.

Published online: 2023-12

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  • 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.

  • Keywords

Quantum Zakharov system, Subsonic limit, Compact finite difference method, Uniformly accurate, Error estimate.

  • AMS Subject Headings

35Q55, 65M06, 65M12, 65M15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-289, author = {Zhang , Gengen and Su , Chunmei}, title = {Uniform Error Bounds of a Conservative Compact Finite Difference Method for the Quantum Zakharov System in the Subsonic Limit Regime}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {42}, number = {1}, pages = {289--312}, 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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2204-m2022-0001}, url = {http://global-sci.org/intro/article_detail/jcm/22161.html} }
TY - JOUR T1 - Uniform Error Bounds of a Conservative Compact Finite Difference Method for the Quantum Zakharov System in the Subsonic Limit Regime AU - Zhang , Gengen AU - Su , Chunmei JO - Journal of Computational Mathematics VL - 1 SP - 289 EP - 312 PY - 2023 DA - 2023/12 SN - 42 DO - http://doi.org/10.4208/jcm.2204-m2022-0001 UR - https://global-sci.org/intro/article_detail/jcm/22161.html KW - Quantum Zakharov system, Subsonic limit, Compact finite difference method, Uniformly accurate, Error estimate. AB -

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.

Gengen Zhang & Chunmei Su. (2023). Uniform Error Bounds of a Conservative Compact Finite Difference Method for the Quantum Zakharov System in the Subsonic Limit Regime. Journal of Computational Mathematics. 42 (1). 289-312. doi:10.4208/jcm.2204-m2022-0001
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