A fully multidimensional positive definite advection scheme with small implicit diffusion

The idea of the simple positive d finite advection scheme presented previously inMonthly Weather Review (1 I I (1983), 479) is improved for an optional multidimensional caseand is presented ina generalized format. The accuracy of the scheme is discussed and a review of existing options is presented and illustrated through numerical tests. 1. INTR~~LJCTI~N In numerical modeling ofphysical phenomena it is often necessary to solve the advective transport equation for positive definite scalar functions. Numerical schemes of second- orhigher-order accuracy an produce n gative values inthe solution due to the dispersive ripples. Lower-order schemes, uch as the donor cell or Lax-Friedrichs, or igher-order schemes with zeroth-order diffusion added produce no ripples but suffer from excessive mplicit d ffusion. In the last ten years a possible resolution of this dilemma has been developed in the form of hybrid schemes, in which the advective fluxes are given as a weighted average ofa first-order positive definite scheme’s fluxes and a higher-order scheme’s fluxes. The difference i d ter-mination fthe weights inthe calculation of the average advective fluxes has led t