Kinematic dynamos using constrained transport with high order Godunov
schemes and adaptive mesh refinement
Romain Teyssiera,b,
Sebastien Fromangc,
Emmanuel Dormyd,e,b
a CEA/DSM/DAPNIA/Service d'Astrophysique, Gif-sur-Yvette,
91191 Cedex, France
b Institut d'Astrophysique de Paris, 98bis Bd Arago, 75014 Paris, France
c Astronomy Unit, Queen Mary, University of London, Mile End Road, London E1 4NS, UK
d Laboratoire de Physique Statistique, E.N.S., 24, rue Lhomond, 75231 Paris Cedex 05, France
e Departement de Geomagnetisme, I.P.G.P./C.N.R.S., 4 Place Jussieu, 75252
Paris Cedex 05, France
Journal of Computational Physics,
Volume 218, Issue 1 , 10 October 2006, Pages 44-67
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
Òfinite-surfaceÓ scheme for the induction equation. This naturally leads to
the well-known Òconstrained transportÓ 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 RungeÐKutta),
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.
Keywords: Magnetohydrodynamics and electrohydrodynamics;
Hydrodynamic and hydromagnetic problems; Finite difference methods;
Induction equation; Magnetohydrodynamics; Godunov scheme; Adaptive mesh
refinement; Numerical schemes