  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.
