Local adaptive mesh refinement for shock hydrodynamics
Journal of Computational Physics
Multidimensional upwind methods for hyperbolic conservation laws
Journal of Computational Physics
Fully threaded tree algorithms for adaptive refinement fluid dynamics simulations
Journal of Computational Physics
Journal of Computational Physics
A solution-adaptive upwind scheme for ideal magnetohydrodynamics
Journal of Computational Physics
The &Dgr; • = 0 constraint in shock-capturing magnetohydrodynamics codes
Journal of Computational Physics
Divergence-free adaptive mesh refinement for Magnetohydrodynamics
Journal of Computational Physics
Hyperbolic divergence cleaning for the MHD equations
Journal of Computational Physics
Divergence- and curl-preserving prolongation and restriction formulas
Journal of Computational Physics
Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries
Journal of Computational Physics
Journal of Computational Physics
A novel approach of divergence-free reconstruction for adaptive mesh refinement
Journal of Computational Physics
An unsplit, cell-centered Godunov method for ideal MHD
Journal of Computational Physics
An unsplit Godunov method for ideal MHD via constrained transport
Journal of Computational Physics
Finite element form of FDV for widely varying flowfields
Journal of Computational Physics
Hi-index | 31.45 |
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.