On the quantification of numerical dissipation and dispersion in CFD

The method developed by Schranner et al. (2015) allows to estimate the numerical dissipation through a kinetic energy balance equation averaged over sub-domains and was applied with success to the Navier-Stokes solvers for compressible and incompressible flows. In this work we show that the method can be generalized to other PDEs and that for the linear advection equation it is in agreement with the modified equation analysis. Novelty of this work is the extension of the original method to the estimation of the dispersive error. The extension is based on a split of the residual of the kinetic energy balance equation that allows to estimate both dissipative and dispersive coefficients through a least squares method. The procedure is validated on the linear advection equation for several numerical schemes for which dispersive and dissipative errors are known. When the new method is applied to non-linear PDEs the estimates of the numerical dissipation obtained using the original method are recovered. The rigorous results obtained in this work further support the previous heuristic method for estimating numerical errors in the course of simulations performed with arbitrary Navier-Stokes solvers.