A 2D Fourth-Order CESE Scheme for Solving the Ideal MHD Equations

We construct a two-dimensional fourth-order space-time conservation element and solution element (CESE) schemes for solving the ideal magnetohydrodynamics (MHD) equations. In the CESE scheme, the flow variables are calculated by using the same procedure as that of the original second-order CESE scheme. The scheme preserves most favorable attributes of the original second-order CESE method. Moreover, it is simple and easy to program. The numerical example for the smooth Alfvén wave problem suggests that the scheme can achieve the fourth-order accuracy for smooth solutions. In order to verify the efficiency of the schemes, we simulate several 2D MHD problems. We find that the fourth-order scheme can capture shocks and details of complex flow structures very well, and control the magnetic divergence efficiently. Moreover, the scheme is essentially CFL number insensitive schemes. The last several complex test problems further verify the performance of proposed scheme.