A Fully Multidimensional Positive Definite Advection Transport Algorithm with Small Implicit Diffusion

In numerical modeling of physical phenomena it is often necessary to solve the advective transport equation for positive definite scalar functions. Numerical schemes of secondor higher-order accuracy can produce negative values in the solution due to the dispersive ripples. Lower-order schemes, such as the donor cell or Lax-Friedrichs, or higher-order schemes with zeroth-order diffusion added produce no ripples but suffer from excessive implicit diffusion. 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 of a first-order positive definite scheme’s fluxes and a higher-order scheme’s fluxes. The difference in determination of the weights in the calculation of the average advective fluxes has led to different hybrid schemes. Two main hybrid-type schemes have been developed. One, the so called flux-corrected transport (FCT) method, was originated by Boris and Book [ I-31 and generalized by Zalesak [ 141; the other was developed by Harten and Zwas [ 6,9] in the form of the self-adjusting hybrid schemes (SAHS) method. Both methods were constructed to deal effectively with shocks and contact discontinuities. Solutions of the advection transport equation obtained by using FCT or SAHS maintain positive definiteness of the initial condition and, as can be seen from presented tests (Zalesak [ 141, Harten [6]), be very accurate. Unfortunately, application of these methods to the modeling of complex multidimensional hydrodynamical systems like atmospheric phenomena is rather limited due to the excessive computer time required. Furthermore, in many hydrodynamical systems,