Multidimensional Flux-Form Semi-Lagrangian Transport Schemes

TLDR

The authors propose an algorithm that extends one‑dimensional, forward‑in‑time, upstream‑biased flux‑form transport schemes to multidimensions and enables arbitrarily long time steps. This flux‑form semi‑Lagrangian (FFSL) scheme, implementable efficiently on the sphere, is derived from the semi‑Lagrangian extension of time steps and its stability is examined analytically and empirically. Idealized and realistic three‑dimensional global transport simulations demonstrate that the algorithm is conservative, upstream‑biased, monotonicity‑constrained, and preserves tracer correlations, accurately reflecting physical transport characteristics.

Abstract

An algorithm for extending one-dimensional, forward-in-time, upstream-biased, flux-form transport schemes (e.g., the van Leer scheme and the piecewise parabolic method) to multidimensions is proposed. A method is also proposed to extend the resulting Eulerian multidimensional flux-form scheme to arbitrarily long time steps. Because of similarities to the semi-Lagrangian approach of extending time steps, the scheme is called flux-form semi-Lagrangian (FFSL). The FFSL scheme can be easily and efficiently implemented on the sphere. Idealized tests as well as realistic three-dimensional global transport simulations using winds from data assimilation systems are demonstrated. Stability is analyzed with a von Neuman approach as well as empirically on the 2D Cartesian plane. The resulting algorithm is conservative and upstream biased. In addition, it contains monotonicity constraints and conserves tracer correlations, therefore representing the physical characteristics of constituent transport.