A high order Godunov scheme with constrained transport and adaptive mesh refinement for astrophysical and geophysical MHD

We propose to extend the well-known MUSCIL-Hancock scheme for Euler equations to the induction equation modeling the magnetic field evolution ill kinematic dynamo problems. The scheme is based oil in 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 oil the magnetic field representation. The second ingredient in the MUSCIL 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 oil this predictor step, showing that three possible variants can be considered. We show that the most aggressive formulations reach the same level of accuracy than the other ones, at it lower computational cost. More interestingly, these 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. The scheme is then adaptated to solve for the full MHD equations using the same methodology. Through a series of test problems, we illustrate the performances of this new code using two different MHD Riemann solvers (La x-Fried rich Roe) and the need of the Adaptive Mesh Refinement capabilities in some cases. and Finally, we show its versatility by applying it to the ABC dynamo problem and to the collapse of a magnetized cloud core.