Unstaggered central schemes with constrained transport treatment for ideal and shallow water magnetohydrodynamics

Abstract

We propose an unstaggered, non-oscillatory, second-order accurate central scheme for approximating the solution of general hyperbolic systems in one and two space dimensions, and in particular for estimating the solution of ideal magnetohydrodynamic problems and shallow water magnetohydrodynamic problems. In contrast with standard central schemes that evolve the numerical solution on two staggered grids at consecutive time steps, the method we propose evolves the numerical solution on a single unique grid, and avoids the resolution of the Riemann problems arising at the cell interfaces thanks to a layer of ghost staggered cells implicitly used while updating the numerical solution on the control cells. To satisfy the divergence-free constraint of the magnetic field/flux in the numerical solution of ideal/shallow water magnetohydrodynamic problems, we adapt Evans and Hawley's constrained transport method to our unstaggered base scheme and use it to correct the magnetic field/flux components at the end of each time step. The resulting method is used to solve classical ideal/shallow water magnetohydrodynamic problems; the obtained results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the efficiency and the potential of the proposed method.