The forward-in-time upstream advection scheme:extension to higher orders
Monthly Weather Review
Flux forms of these schemes are also derived with Crowley's methodology. It is shown that these 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 2D using time splitting. 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 6th-order schemes appear to be the best balance between efficiency and accuracy.
C. Tremback, J. Powell, W. Cotton, R. Pielke. “The forward-in-time upstream advection scheme: extension to higher orders,” Monthly Weather Rev. 115, 540–555, 1987.