Numerical equivalence of advection in flux and advective forms and quadratically conservative high-order advection schemes

Abstract

In finite-difference representations of the conservation equations, the flux form of the advection terms is often preferred to the advective form because of the immediate conservation of advected quantity. The scheme can be designed to further conserve higher-order moments, for example, the kinetic energy, which is important to the suppression of nonlinear instability. It is pointed out here that in most cases an advective form that is numerically equivalent to the flux form can be found, for schemes based on centered difference. This is also true for higher-order schemes and is not restricted to a particular grid type. An advection scheme that is fourth-order accurate in space for uniform advective flows is proposed that conserves both first and second moments of the advected variable in a nonhydrostatic framework. The role of the elastic correction term in addition to the pure flux term in compressible nonhydrostatic models is also discussed.