Monotone Flux Limitation in the Area-preserving Flux-form Advection Algorithm

Abstract

The area-preserving flux-form advection algorithm is extended to monotonicity. For this, the nonlinear positive-definite flux limitation of the original approach is replaced by new monotone flux limiters. The monotone fluxes are derived for one-dimensional constant transport velocities. The deformation occurring in divergent flow is accounted for by adding to the monotone advection fluxes a correction term, which has been derived from the deformation of the upstream method. The final algorithm is applicable to arbitrary multidimensional transport problems. However, due to the use of the time-splitting method, it is strictly monotone only in uniform flow fields. Results of different one- and two-dimensional advection experiments are presented, demonstrating that the monotone flux limitation is an attractive alternative to the positive-definite algorithm. Amplitude and phase speed errors are somewhat larger in the monotone advection scheme. The computational effort of the new version is not much larger than that of the positive definite scheme. Thus, it is concluded that for many applications of atmospheric modeling, the monotone area-preserving flux-form advection algorithm is an accurate and numerically efficient method for the solution of the transport equation.