Entropy estimates for a class of schemes for the euler equations
This work provides theoretical entropy stability guarantees for specific numerical schemes for the Euler equations, which is important for computational fluid dynamics but is incremental as it extends known entropy analysis to a particular class of schemes.
The paper derives entropy estimates for a class of schemes for the Euler equations based on the internal energy equation with upwinding by material velocity. The implicit first-order upwind scheme satisfies a local entropy inequality, while generalized schemes satisfy a weaker property with a remainder term that tends to zero under L∞ and BV control.
In this paper, we derive entropy estimates for a class of schemes for the Euler equations which present the following features: they are based on the internal energy equation (eventually with a positive corrective term at the righ-hand-side so as to ensure consistency) and the possible upwinding is performed with respect to the material velocity only. The implicit-in-time first-order upwind scheme satisfies a local entropy inequality. A generalization of the convection term is then introduced, which allows to limit the scheme diffusion while ensuring a weaker property: the entropy inequality is satisfied up to a remainder term which is shown to tend to zero with the space and time steps, if the discrete solution is controlled in L $\infty$ and BV norms. The explicit upwind variant also satisfies such a weaker property, at the price of an estimate for the velocity which could be derived from the introduction of a new stabilization term in the momentum balance. Still for the explicit scheme, with the above-mentioned generalization of the convection operator, the same result only holds if the ratio of the time to the space step tends to zero.