The Forward–in-Time Upstream Advection Scheme: Extension to Higher Orders

A generalized forward-in-time upstream advective operator following the methodology of Crowley is developed and advective schemes of orders 1 through 10 are tested analytically and numerically. Flux forms of these schemes are also derived with Crowley’s methodology. It is shown that thew flux forms do not reduce to the advective form with constant velocity and grid spacing for schemes of order 3 and higher and they are not as accurate as the advective form. A new flux form is derived which does reduce to the advective form under the conditions of constant velocity and grid spacing. The schemes were tested in two dimensions using time splitting. In the rotating cone test, the advective and new flux forms performed identically, while the other flux form had larger dissipation and dispersion errors. In the deformational flow field test, the advective forms were unstable for both time steps tested. Use flux forms were less unstable for the higher Courant number and the domain as a whole was stable for the lower Courant number. With respect to order of the various forms, the schemes performed consistent with the linear stability analysis; errors decrease as the order of the scheme becomes greater. The sixth-order schemes appear to be the best balance between efficiency and accuracy.