## Abstract

We propose to extend the well-known MUSCL-Hancock scheme for Euler equations to the induction equation modeling the magnetic field evolution in kinematic dynamo problems. The scheme is based on an integral form of the underlying conservation law which, in our formulation, results in a "finite-surface" scheme for the induction equation. This naturally leads to the well-known "constrained transport" method, with additional continuity requirement on the magnetic field representation. The second ingredient in the MUSCL scheme is the predictor step that ensures second order accuracy both in space and time. We explore specific constraints that the mathematical properties of the induction equations place on this predictor step, showing that three possible variants can be considered. We show that the most aggressive formulations (referred to as C-MUSCL and U-MUSCL) reach the same level of accuracy as the other one (referred to as Runge-Kutta), at a lower computational cost. More interestingly, these two schemes are compatible with the adaptive mesh refinement (AMR) framework. It has been implemented in the AMR code RAMSES. It offers a novel and efficient implementation of a second order scheme for the induction equation. We have tested it by solving two kinematic dynamo problems in the low diffusion limit. The construction of this scheme for the induction equation constitutes a step towards solving the full MHD set of equations using an extension of our current methodology.

Original language | English (US) |
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Pages (from-to) | 44-67 |

Number of pages | 24 |

Journal | Journal of Computational Physics |

Volume | 218 |

Issue number | 1 |

DOIs | |

State | Published - Oct 10 2006 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Numerical Analysis
- Modeling and Simulation
- Physics and Astronomy (miscellaneous)
- Physics and Astronomy(all)
- Computer Science Applications
- Computational Mathematics
- Applied Mathematics

## Keywords

- Adaptive mesh refinement
- Finite difference methods
- Godunov scheme
- Hydrodynamic and hydromagnetic problems
- Induction equation
- Magnetohydrodynamics
- Magnetohydrodynamics and electrohydrodynamics
- Numerical schemes