### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 965-993 |

Number of pages | 29 |

Journal | ESAIM: Mathematical Modelling and Numerical Analysis |

Volume | 50 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2016 |

### Fingerprint

### Keywords

- Constrained transport
- Discontinuous galerkin
- Divergence-free
- Error estimates
- Ideal magnetohydrodynamic (MHD) equations
- Magnetic induction equations
- Stability

### ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Modeling and Simulation
- Applied Mathematics

### Cite this

**Stability analysis and error estimates of an exactly divergence-free method for the magnetic induction equations.** / Yang, He; Li, Fengyan.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Stability analysis and error estimates of an exactly divergence-free method for the magnetic induction equations

AU - Yang, He

AU - Li, Fengyan

PY - 2016/1/1

Y1 - 2016/1/1

N2 - 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.

AB - 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.

KW - Constrained transport

KW - Discontinuous galerkin

KW - Divergence-free

KW - Error estimates

KW - Ideal magnetohydrodynamic (MHD) equations

KW - Magnetic induction equations

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=84975257523&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84975257523&partnerID=8YFLogxK

U2 - 10.1051/m2an/2015061

DO - 10.1051/m2an/2015061

M3 - Article

VL - 50

SP - 965

EP - 993

JO - ESAIM: Mathematical Modelling and Numerical Analysis

JF - ESAIM: Mathematical Modelling and Numerical Analysis

SN - 0764-583X

IS - 4

ER -