High Order Finite Difference WENO Schemes for Ideal Magnetohydrodynamics

In this work, we focus on the development of high-order numerical methods for the ideal magnetohydrodynamic (MHD) equations. Following the idea in [1], a constrained transport approach is employed to control divergence errors in the magnetic field, which will give us a non-conservative weakly hyperbolic system. The finite difference (FD) weighted essentially non-oscillatory (WENO) scheme for Hamilton-Jacobi (HJ) equations with artificial resistivity was used to solve this system. In order to avoid the artificial diffusion and reduce error, in this work, we would like to use the scheme in [2] to solve the HJ equation, which relies on a special kernel-based formulation of the solutions and successive convolution. A high order WENO integration and a nonlinear filter will help us to capture the correct viscosity solution. Moreover, a positivity-preserving property is used as a limiter. The method is applied to several test problems to verify the order of accuracy on smooth test problems and to show high-resolution on test problems that involve shocks. Here, we will focus on the 2D case, and solve the 3D problems in the future.