Incremental Remapping as a Transport/Advection Algorithm

Abstract

There are two fundamentally different strategies for solving the standard transport or continuity equation, corresponding to whether it is expressed as a partial differential equation or as an integral statement of conservation. The more common approach is to discretize the partial differential equation and to march the solution forward in time. The alternative method is to project cell volumes along Lagrangian trajectories as far forward or backward in time as desired, and then to remap the resulting density distribution onto some target mesh. This latter approach is known as remapping. Remapping has many advantages, not the least of which is that the time step is limited only by accuracy considerations, but it tends to be expensive and complex. In this paper we show that if the time step is made sufficiently short such that trajectories are confined to the nearest neighbor cells, then the remapping may be written as a flux-form transport algorithm, and it becomes nearly as simple and efficient as standard transport schemes. The resulting method, called incremental remapping, retains most of the advantages of general remapping. These include: (a) geometric basis for transport, (b) compatibility of associated tracer transport with simple tracer advection, i.e., retention of tracer monotonicity properties, and (c) efficient handling of multiple tracers since each additional tracer adds only a relatively small incremental cost.