High order central scheme on overlapping cells for magneto-hydrodynamic flows with and without constrained transport method

Abstract

This paper extends the central finite-volume schemes of Liu et al. [Y. Liu, C.-W. Shu, E. Tadmor, M. Zhang, Non-oscillatory hierarchical reconstruction for central and finite-volume schemes, Commun. Comput. Phys. 2 (2007) 933–963] on overlapping cells to the magneto-hydrodynamic (MHD) equations. In particular, we propose a high order divergence-free reconstruction for the magnetic field that uses the face-centered values. We also advance the magnetic field with a high order constrained transport (CT) scheme to preserve the divergence-free condition to machine round-off error. The overlapping cells are natural to be used to calculate the electric field flux without an averaging procedure. We have developed a third-order scheme which is verified by the numerical experiments. Other higher order schemes can be constructed accordingly. Our central constrained transport schemes do not need characteristic decomposition, and are easy to code and combine with un-split discretization of the source and parabolic terms. The overlapping cell representation of the solution is also used to develop more compact reconstruction and less dissipative schemes. The high resolution is achieved by non-oscillatory hierarchical reconstruction, which does not require characteristic decomposition either. The numerical comparisons show that the central schemes with non-CT perform as well as with CT for most of problems. Numerical examples are given to demonstrate efficacy of the new schemes.