A simple locally divergence-free OEDG method for ideal compressible MHD equations

Simulating the ideal compressible MHD equations is challenging, as solutions must remain divergence-free in general and oscillation-free near discontinuities. To address these challenges, we introduce a locally divergence-free, oscillation-eliminating discontinuous Galerkin (LDF-OEDG) method for the ideal compressible MHD equations. In this method, a strong stability-preserving Runge–Kutta scheme is employed to advance the numerical solution in time. At each Runge–Kutta stage, the solution update is followed by a locally divergence-free (LDF) projection, an oscillation-eliminating (OE) procedure, and a final LDF projection. The LDF projections enforce the divergence-free condition on the magnetic field by projecting the solution onto a local divergence-free space in each element. The OE procedure suppresses spurious oscillations near discontinuities by damping the modal coefficients. The first LDF projection ensures a divergence-free solution for computing the damping operator, while the second enforces the divergence-free condition on the final stage solution. Both the OE procedure and the LDF projection are fully decoupled from the Runge–Kutta update and can be seamlessly incorporated into existing DG codes as independent modules. We prove, for the first time, the equivalence between the LDF-OEDG method and the OEDG method constructed with LDF bases. Numerical results for benchmark cases demonstrate the high-order accuracy, strong shock-capturing capability, and robustness of the LDF-OEDG method.