We develop and analyse explicit-in-time schemes for the computation of compressible
flows, based on staggered in space unstructured discretization. Upwinding is performed equation by equation only with respect to the velocity (like in the AUSM family of schemes). The pressure gradient is built as the transpose of the natural divergence, which yields a centered discretization of this term.
In a first time, we address the barotropic Euler equations. The velocity convection term is built in such a way that we are able to derive a discrete kinetic energy balance, with (at the left-hand side) residual terms which are non-negative under a CFL condition. We then show that, in one space dimension, the scheme is consistent in the sense that, if a sequence of discrete solutions converges to some limit, then this limit is a weak entropy solution to the continuous problem. Numerical tests allow to check the convergence of the scheme, and show in addition an approximatively
first-order convergence rate.
We then turn to the full (i.e. non-barotropic) Euler equations. We chose here to solve the internal energy balance instead of the total energy equation, which presents two advantages: first, we don’t need a discretization of this latter quantity, which is rather unnatural since the velocity and the scalar unknowns are not approximated on the same mesh; second, an ad hoc discretization of the internal energy balance ensures its positivity. We show that, under CFL-like conditions, the density and internal energy are kept positive, and the total (i.e. integrated over the whole computational domain) energy cannot grow. The difficult point is to obtain consistency. Indeed, a scheme using the internal energy equation may not converge to a weak solution of the original
system in the presence of shocks. This problem is healed by the following strategy:
1. Establish a kinetic energy identity at the discrete level (with some source terms).
2. Choose source term of the internal energy equation such that the total energy balance is recovered when the mesh and time steps tend to zero.
More precisely speaking, we prove the following theoretical result. In 1D, if we assume the L∞ and BV-stability and the convergence of the scheme, passing to the limit of the discrete kinetic and discrete elastic potential equations, we show that the limit of the sequence of solutions indeed is a weak solution. This result is supported by numerical tests.
Finally, we consider the computation of radial flows, governed by Euler equations in axisymetrical (2D) or spherical (3D) coordinates, and obtain similar results to the previous sections.