Convergence rate estimates are obtained for singular limits of the compressible ideal magnetohydrodynamics equations, in which the Mach and Alfvén numbers tend to zero at different rates. The proofs use a detailed analysis of exact and approximate fast, intermediate, and slow modes together with improved estimates for the solutions and their time derivatives, and the time-integration method. When the small parameters are related by a power law the convergence rates are positive powers of the Mach number, with the power varying depending on the component and the norm. Exceptionally, the convergence rate for two components involve the ratio of the two parameters, and that rate is proven to be sharp via corrector terms. Moreover, the convergence rates for the case of a power-law relation between the small parameters tend to the two-scale convergence rate as the power tends to one. These results demonstrate that the issue of convergence rates for three-scale singular limits, which was not addressed in the authors’ previous paper, is much more complicated than for the classical two-scale singular limits.
Accepté le :
Publié le :
DOI : 10.1051/m2an/2020051
Keywords: Rate of convergence, singular limit, magnetohydrodynamics
@article{M2AN_2021__55_S1_S733_0,
author = {Cheng, Bin and Ju, Qiangchang and Schochet, Steve},
title = {Convergence rate estimates for the low {Mach} and {Alfv\'en} number three-scale singular limit of compressible ideal magnetohydrodynamics},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {S733--S759},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {Suppl\'ement},
doi = {10.1051/m2an/2020051},
mrnumber = {4221327},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2020051/}
}
TY - JOUR AU - Cheng, Bin AU - Ju, Qiangchang AU - Schochet, Steve TI - Convergence rate estimates for the low Mach and Alfvén number three-scale singular limit of compressible ideal magnetohydrodynamics JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - S733 EP - S759 VL - 55 IS - Supplément PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2020051/ DO - 10.1051/m2an/2020051 LA - en ID - M2AN_2021__55_S1_S733_0 ER -
%0 Journal Article %A Cheng, Bin %A Ju, Qiangchang %A Schochet, Steve %T Convergence rate estimates for the low Mach and Alfvén number three-scale singular limit of compressible ideal magnetohydrodynamics %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P S733-S759 %V 55 %N Supplément %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2020051/ %R 10.1051/m2an/2020051 %G en %F M2AN_2021__55_S1_S733_0
Cheng, Bin; Ju, Qiangchang; Schochet, Steve. Convergence rate estimates for the low Mach and Alfvén number three-scale singular limit of compressible ideal magnetohydrodynamics. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S733-S759. doi: 10.1051/m2an/2020051
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