Abstract
Several authors have improved the stability or conservation accuracy of the highly popular second-order Crowley advective differencing scheme in multidimensional flow. However, these improved versions still have the marked dispersive phase lag of the original combined or time-split Crowley scheme. This leads to poor shape preservation of a numerically advected disturbance, and anomalous oscillations upstream of it.
This paper presents a method for reducing the dispersive phase lag in the Dukowicz–Ramshaw and Smolarkiewicz versions of the combined multidimensional Crowley scheme, here designated as Types I and II respectively. The method involves adding upstream-biased corrections for third-order spatial truncation error, analogously termed Types I and II. This approach preserves the one-step explicit nature of the schemes.
The effects of the third-order space correction terms on stability and phase error are evaluated in two dimensions, using linear Fourier component analyses and numerical experiments that advect a cone-shaped scalar distribution in steady nondivergent flows.
Our main findings are: (i) our Type I correction dramatically improves the phase accuracy of the Type I scheme, with only slightly less amplitude preservation or stability; (ii) the Type I scheme also performs much better with our Type I correction than with the Hill third-order correction, which is three times as large with excessive damping and large dispersive phase lead; (iii) despite a larger stability region than its Type I counterpart as emphasized by Smolarkiewicz, the Type II scheme has greater dispersive phase lag and faster damping; (iv) applying the Type I correction to the Type II scheme provides wider stability but much less effective phase error reduction than for the Type I scheme; (v) the phase accuracy of the Type II scheme is dramatically improved by the Type II corrector, but with a highly restrictive stability criterion.
