Conservative and oscillation‐less atmospheric transport schemes based on rational functions

Abstract

In this paper, we present two practical schemes for advection transport equation. The schemes, namely Conservative Semi‐Lagrangian with Rational function (CSLR0) and CSLR1, are two new variants of the Constrained Interpolation Profile‐Conservative Semi‐Lagrangian (CIP‐CSL) type methods [Yabe et al., 2001]. In these schemes, the subgrid profile is approximated within a single cell by rational interpolation functions. Both the cell‐integrated average and the values at the two cell interfaces are employed for the interpolation construction. The interface value is computed through a semi‐Lagrangian procedure, while the cell‐integrated quantity is advanced via a flux form formulation that is equivalent to a mass conservative remapping. The schemes are exactly conservative with regards to the cell integration of the advected quantity and have quite small dispersion and diffusion errors. The rational interpolations suppress spurious oscillation, and a continuous profile with a smoothness of at least C0 can be obtained over the whole computational domain. Without any explicit computation of flux or slope limiter, the schemes are more computationally efficient than the piecewise parabolic method (PPM) scheme in the context of 1D. Our numerical tests show that the presented schemes are comparable to the PPM method with regards to numerical dispersion, clipping, and shape preserving.