Use of the Harmonic Mean in the Discretization of the Compressible Flow Equations

When solving the compressible Navier–Stokes equations, the specification of numerical fluxes for the primary variables is often required. To do that, the values of the variables at the cell faces—where the fluxes are computed—have to be evaluated through an appropriate interpolation of the nodal values. This operation is most commonly carried out through an arithmetic mean of adjacent values; however alternative means can be employed, especially in the context of structure-preserving discretizations. These numerical methods are designed to satisfy some physics-compatible constraints, such as the discrete enforcement of the induced balance of secondary quantities of interest.For example, the geometric mean can be used to correctly evolve square-root variables while the logarithmic mean appears in schemes that aim at exact entropy conservation. This work presents an additional lesser-known mean that can be of interest: the harmonic mean. It has been recently shown that its use as an interpolator of the internal energy is at the base of asymptotically entropy conserving schemes, able to arbitrarily reduce the numerical error on entropy production due to the spatial discretization. Moreover, the harmonic mean can be shown to have an important role also regarding the pressure equilibrium preservation property which can be useful to avoid spurious pressure oscillations.