  Numerical simulations of ideal compressible magnetohydrodynamic (MHD)
equations are challenging, as the solutions are required to be magnetic
divergence-free for general cases as well as oscillation-free for cases
involving discontinuities. To overcome these difficulties, we develop a locally
divergence-free oscillation-eliminating discontinuous Galerkin (LDF-OEDG)
method for ideal compressible MHD equations. In the LDF-OEDG method, the
numerical solution is advanced in time by using a strong stability preserving
Runge-Kutta scheme. Following the solution update in each Runge-Kutta stage, an
oscillation-eliminating (OE) procedure is performed to suppress spurious
oscillations near discontinuities by damping the modal coefficients of the
numerical solution. Subsequently, on each element, the magnetic filed of the
oscillation-free DG solution is projected onto a local divergence-free space,
to satisfy the divergence-free condition. The OE procedure and the LDF
projection are fully decoupled from the Runge-Kutta stage update, and can be
non-intrusively integrated into existing DG codes as independent modules. The
damping equation of the OE procedure can be solved exactly, making the LDF-OEDG
method remain stable under normal CFL conditions. These features enable a
straightforward implementation of a high-order LDF-OEDG solver, which can be
used to efficiently simulate the ideal compressible MHD equations. Numerical
results for benchmark cases demonstrate the high-order accuracy, strong shock
capturing capability and robustness of the LDF-OEDG method.
