A new locally divergence-free path-conservative central-upwind scheme for ideal and shallow water magnetohydrodynamics (Q6562380)

The authors develop a new second-order unstaggered path-conservative central-upwind (PCCU) scheme for the ideal and shallow water magnetohydrodynamic (MHD) systems. This scheme is (i) locally divergence-free; (ii) Riemann-problem-solver-free; (iii) high-resolution; (iv) robust; and (v) nonoscillatory. The derivation of the scheme is based on the Godunov-Powell nonconservative modifications of the studied MHD systems. The local divergence-free property is enforced by augmenting the studied systems with the evolution equations for the corresponding derivatives of the magnetic field components and by using these evolved quantities in the design of a special piecewise linear reconstruction of the magnetic field, which also guarantees a nonoscillatory nature of the resulting scheme. Several numerical tests are presented, for both ideal and shallow water MHD systems, to show the robustness of the proposed scheme and its ability not only to achieve high resolution, but also to preserve the positivity of computed quantities such as density, pressure, and water depth.