### Non-Oscillatory Central Schemes for One- and Two-Dimensional MHD Equations. II High-Order Semi-Discrete Schemes

Jorge Balbas and Eitan Tadmor

Abstract: We present a new family of high-resolution, non-oscillatory semi-discrete central schemes for the approximate solution of the ideal Magnetohydrodynamics (MHD) equations. This is the second part of our work, where we are passing from the fully-discrete staggered schemes in {BTW}[J. Balb{\'a}s, E. Tadmor, and C.-C. Wu, Non-oscillatory central schemes for one- and two-dimensional {MHD} equations. {I}. JCP, 201(1):261--285, 2004]. to the semi-discrete formulation advocated in [A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. JCP, 160(1):241--282, 2000]. This semi-discrete formulation retains the simplicity of fully-discrete central schemes while enhancing efficiency and adding versatility. The semi-discrete algorithm offers a wider range of options to implement its two key steps: non-oscillatory reconstruction of point values followed by the evolution of the corresponding point valued fluxes. We present the solution of several prototype MHD problems. Solutions of one-dimensional Brio-Wu shock-tube problems and the two-dimensional Kelvin-Helmholtz instability, Orszag-Tang vortex system, and the disruption of a high density cloud by a strong shock are carried out using third- and fourth-order central schemes based on the CWENO reconstructions. These results complement those presented in \cite{BTW} and confirm the remarkable versatility and simplicity of central schemes as black-box, Jacobian-free MHD solvers. Furthermore, our numerical experiments demonstrate that this family of semi-discrete central schemes preserves the $\nabla \cdot {\mathbf B} = 0$-constraint within machine round-off error; happily, no constrained-transport enforcement is needed.

Paper:
Available as PDF (5.1 Mbytes).
Author(s):
Jorge Balbas, email: jbalbas at umich dot edu
Eitan Tadmor, email: tadmor at cscamm dot umd dot edu
Publishing information:
SIAM Journal on Scientific Computing