Nonoscillatory Central Schemes for One- and Two-Dimensional Magnetohydrodynamics Equations. II: High-Order SemiDiscrete Schemes

Abstract

We present a new family of high-resolution, nonoscillatory semidiscrete 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 [J. Balbás, E. Tadmor, and C.-C. Wu, J. Comput. Phys., 201 (2004), pp. 261-285] to the semidiscrete formulation advocated in [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 241-282]. This semidiscrete formulation retains the simplicity of fully discrete central schemes while enhancing efficiency and adding versatility. The semidiscrete algorithm offers a wider range of options to implement its two key steps: nonoscillatory reconstruction of point values followed by the evolution of thecorresponding 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 central WENO reconstructions. These results complement those presented in our earlier work 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 semidiscrete central schemes preserves the $\nabla \cdot {\bf B} = 0$-constraint within machine round-off error; happily, no constrained-transport enforcement is needed.