Central finite volume schemes with constrained transport divergence treatment for three-dimensional ideal MHD

Abstract

We present three-dimensional central finite volume methods for solving systems of hyperbolic equations. Based on the Lax–Friedrichs and Nessyahu–Tadmor one-dimensional central finite difference schemes, the numerical methods we propose involve an original and a staggered grid in order to avoid the resolution of the Riemann problems at the cell interfaces. The cells of the original grid are Cartesian (cubes with faces parallel to the axes) while those of the staggered grid are either Cartesian or diamond-shaped. We apply these methods and solve some ideal magnetohydrodynamics problems. To satisfy the solenoidal property of the magnetic field in the numerical solution, we present an adaptation of Evans and Hawley’s constrained transport method for central schemes which we call “CTCS”. The CTCS method is easy to implement, it deals directly with the computed solution and does not require any additional staggering for the magnetic field components; furthermore, it preserves the second-order accuracy of the base scheme. Even without the application of the CTCS procedure, our numerical base schemes do not break down, and may even in some cases deliver reasonable results. The diamond dual cell scheme has a slight advantage for shocks and contact discontinuities. Our numerical results are in good agreement with corresponding results appearing in the recent literature.