Non-oscillatory relaxation schemes for one-dimensional ideal magnetohydrodynamic equations

We present a high-order accurate relaxed non-oscillatory scheme for solving magnetohydrodynamic (MHD) equations. The computations reported here demonstrate the remarkable simplicity and versatility of semi-discrete relaxation schemes as solvers for ideal MHD equations. Simulations will be presented for two prototype MHD problems, the one-dimensional Brio–Wu shock tube problems. A qualitative comparison reveals an excellent agreement with previous results based on other approaches. Furthermore, the numerical results demonstrate that such semi-discrete schemes preserve the ∇⋅B=0 constraint without necessarily needing the constrained-transport enforcement.