Solving the MHD equations by the space-time conservation element and solution element method

We apply the Space-Time Conservation Element and Solution Element (CESE) method to solve the ideal MHD equations with special emphasis on satisfying the divergence free constraint of magnetic field, i.e., ∇⋅B = 0. In the setting of the CESE method, four approaches are employed: (i) the original CESE method without any additional treatment, (ii) a simple corrector procedure to update the spatial derivatives of magnetic field B after each time marching step to enforce ∇⋅B = 0 at all mesh nodes, (iii) an constraint-transport method by using a special staggered mesh to calculate magnetic field B, and (iv) the projection method by solving a Poisson solver after each time marching step. To demonstrate the capabilities of these methods, two benchmark MHD flows are calculated (i) a rotated one-dimensional MHD shock tube problem, and (ii) a MHD vortex problem. The results show no differences between different approaches and all results compare favorably with previously reported data.