A well balanced scheme on staggered grids

It was a well established fact that a naive treatment of zeroth oder source terms, like gravity forces, in the numerical simulation of conservation laws leads to instable or unphysical solutions.

In a remarkable paper with R. Botchorishvili and B. Perthame, A. Vasseur introduced an elegant way to fix this issue, by setting up a suitable definition of the numerical fluxes which is compatible with the equilibrium solutions of the problems.

We revisit this approach in the framework of staggered discretizations for the Euler equations, where densities and velocities are stored in dual discrete locations.

We derive first and second order versions of a well-balanced scheme, constructed by using the principles of kinetic schemes and hydrostatic reconstructions.