In this paper, we consider an exactly divergence-free scheme to solve the magnetic induction equations. This problem is motivated by the numerical simulations of ideal magnetohydrodynamic (MHD) equations, a nonlinear hyperbolic system with a divergence-free condition on the magnetic field. Computational methods without satisfying such condition may lead to numerical instability. One class of methods, constrained transport schemes, is widely used as divergence-free treatments. So far there is not much analysis available for such schemes. In this work, we take an exactly divergence-free scheme proposed by [Li and Xu, J. Comput. Phys. 231 (2012) 2655–2675] as a candidate of the constrained transport schemes, and adapt it to solve the magnetic induction equations. For the resulting scheme applied to the equations with a constant velocity field, we carry out von Neumann analysis for numerical stability on uniform meshes. We also establish the stability and error estimates based on energy methods. In particular, we identify the stability mechanism due to the spatial and temporal discretizations, and the role of the exactly divergence-free property of the numerical solution for stability. The analysis based on energy methods can be extended to non-uniform meshes, and they can also be applied to the magnetic induction equations with a variable velocity field, which is more relevant to the MHD simulations.

DOI: 10.1051/m2an/2015061

Keywords: Stability, error estimates, ideal magnetohydrodynamic (MHD) equations, constrained transport, divergence-free, discontinuous galerkin, magnetic induction equations

^{1}; Li, Fengyan

^{2}

@article{M2AN_2016__50_4_965_0, author = {Yang, He and Li, Fengyan}, title = {Stability analysis and error estimates of an exactly divergence-free method for the magnetic induction equations}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {965--993}, publisher = {EDP-Sciences}, volume = {50}, number = {4}, year = {2016}, doi = {10.1051/m2an/2015061}, zbl = {1348.78028}, mrnumber = {3521708}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2015061/} }

TY - JOUR AU - Yang, He AU - Li, Fengyan TI - Stability analysis and error estimates of an exactly divergence-free method for the magnetic induction equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 965 EP - 993 VL - 50 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015061/ DO - 10.1051/m2an/2015061 LA - en ID - M2AN_2016__50_4_965_0 ER -

%0 Journal Article %A Yang, He %A Li, Fengyan %T Stability analysis and error estimates of an exactly divergence-free method for the magnetic induction equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 965-993 %V 50 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015061/ %R 10.1051/m2an/2015061 %G en %F M2AN_2016__50_4_965_0

Yang, He; Li, Fengyan. Stability analysis and error estimates of an exactly divergence-free method for the magnetic induction equations. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 50 (2016) no. 4, pp. 965-993. doi : 10.1051/m2an/2015061. http://www.numdam.org/articles/10.1051/m2an/2015061/

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