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In this work we develop a class of high-order finite difference weightedessentially non-oscillatory (FD-WENO) schemes for solving the idealmagnetohydrodynamic (MHD) equations in 2D and 3D. The philosophy of this workis to use efficient high-order WENO spatial discretizations with high-orderstrong stability-preserving Runge-Kutta (SSP-RK) time-stepping schemes.Numerical results have shown that with such methods we are able to resolvesolution structures that are only visible at much higher grid resolutions withlower-order schemes. The key challenge in applying such methods to ideal MHD isto control divergence errors in the magnetic field. We achieve this byaugmenting the base scheme with a novel high-order constrained transportapproach that updates the magnetic vector potential. The predicted magneticfield from the base scheme is replaced by a divergence-free magnetic field thatis obtained from the curl of this magnetic potential. The non-conservativeweakly hyperbolic system that the magnetic vector potential satisfies is solvedusing a version of FD-WENO developed for Hamilton-Jacobi equations. Theresulting numerical method is endowed with several important properties: (1)all quantities, including all components of the magnetic field and magneticpotential, are treated as point values on the same mesh (i.e., there is no meshstaggering); (2) both the spatial and temporal orders of accuracy arefourth-order; (3) no spatial integration or multidimensional reconstructionsare needed in any step; and (4) special limiters in the magnetic vectorpotential update are used to control unphysical oscillations in the magneticfield. Several 2D and 3D numerical examples are presented to verify the orderof accuracy on smooth test problems and to show high-resolution on testproblems that involve shocks.

Finite Difference Weighted Essentially Non-Oscillatory Schemes with Constrained Transport for Ideal Magnetohydrodynamics