On a Class of Structure-Preserving Discretizations in Compressible Flows

A crucial research topic in computational fluid dynamics is that of developing numerical methods able to retain at discrete level some key properties of the continuous governing equations, with the objective of having more accurate and reliable simulations. In order to increase robustness, in recent years various Kinetic Energy Preserving (KEP) schemes have been proposed and their use for the spatial discretization of the mass and momentum equations is now considered a necessity for a reliable simulation. In analogy, KEP schemes have also been applied to the energy equation, preserving quadratic quantities which, however, may not be of physical interest. On the other hand, the preservation of other structural properties of the continuous equations, as the correct evolution of square-root variables, may in some case be more meaningful. As an example, the square root of internal energy is proportional to sound speed. In this paper we investigate a class of discretizations capable of correctly reproducing the evolution of square-root quantities to various forms of the energy equation of the compressible Euler equations for shock-free flows. The resulting schemes have been analyzed and tested on their ability to enhance the correct preservation of additional invariants, such as entropy, using common inviscid tests.