Unstaggered central schemes for hyperbolic systems

Abstract

We develop an unstaggered central scheme for approximating the solution of general two‐dimensional hyperbolic systems. In particular, we are interested in solving applied problems arising in hydrodynamics and astrophysics. 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 grid, and avoids the resolution of the Riemann problems arising at the cell interfaces, thanks to a layer of ghost cells implicitly used. The numerical base scheme is used to solve shallow water equation problems and ideal magnetohydrodynamic problems. To satisfy the divergence‐free constraint of the magnetic field in the numerical solution of ideal magnetohydrodynamic problems, we adapt Evans and Hawley’s the constrained transport method to our unstaggered base scheme, and apply it to correct the magnetic field components at the end of each time step. 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.