Monotone Advection on the Sphere: An Eulerian Versus Semi-Lagrangian Approach

Abstract

The purpose of this paper is twofold. First, a formalism is presented that extends the conceptual framework identified by Ritchie as the “semi-Lagrangian method without interpolation.” While his words for this concept refer to a particular class of semi-Lagrangian approximations, the idea is actually much more general. The formalism may be used to convert any advection algorithm into the semi-Lagrangian format, and it makes most algorithms sufficient for the integration of flows characterized by large Courant numbers. The formalism is presented in an arbitrary curvilinear system of coordinates. Second, exploiting the generality of the theoretical considerations, the formalism is implemented in solving a practical problem of scalar advection in spherical geometry. Rather than elaborating on Ritchie's semi-Lagrangian techniques employing centered-in-time differencing, the focus is on the alternative of forward-in-time, dissipative finite-difference schemes. This class of schemes offers attractive computational properties in terms of the solutions' accuracy and preservation of a sign or monotonicity.