A Positive Definite Advection Scheme Obtained by Nonlinear Renormalization of the Advective Fluxes

Andreas Bott Meteorological Institute, Johannes Gutenberg University, Mainz, FRG

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Abstract

A new method is developed to obtain a conservative and positive definite advection scheme that produces only small numerical diffusion. Advective fluxes are computed utilizing the integrated flux form of Tremback et al. These fluxes are normalized and then limited by upper and lower values. The resulting advection equation is numerically solved by means of the usual upstream procedure. The proposed treatment is not restricted to the integrated flux form but may also be applied to other known advection algorithms which are formulated in terms of advective fluxes.

Different numerical tests are presented illustrating that the proposed scheme strongly reduces numerical and diffusion and simultaneously requires only small computational effort. For Corant numbers with absolute values not exceeding one, the scheme preserves numerical stability except in strong deformational flow fields where slight instabilities may occur.

Abstract

A new method is developed to obtain a conservative and positive definite advection scheme that produces only small numerical diffusion. Advective fluxes are computed utilizing the integrated flux form of Tremback et al. These fluxes are normalized and then limited by upper and lower values. The resulting advection equation is numerically solved by means of the usual upstream procedure. The proposed treatment is not restricted to the integrated flux form but may also be applied to other known advection algorithms which are formulated in terms of advective fluxes.

Different numerical tests are presented illustrating that the proposed scheme strongly reduces numerical and diffusion and simultaneously requires only small computational effort. For Corant numbers with absolute values not exceeding one, the scheme preserves numerical stability except in strong deformational flow fields where slight instabilities may occur.

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