On the estimation of artificial dissipation and dispersion errors in a generic partial differential equation

Abstract

A previously developed method, which allows for the estimation of the numerical dissipation through a kinetic energy balance equation averaged over a sub-domain, was applied with success to Navier-Stokes solvers for compressible and incompressible flows. In this work we show that the method can be generalized to other Partial Differential Equations (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 for the estimation of both dissipative and dispersive coefficients through a least squares regression. 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 linear or non-linear PDEs the estimates of the numerical dissipation obtained using the original method are recovered. The obtained rigorous results further support the previous heuristic method for estimating numerical errors in the course of simulations performed with arbitrary Navier-Stokes solvers.