A Fully Lagrangian Advection Scheme

A numerical method for passive scalar and self-advection dynamics, Lagrangian rearrangement, is proposed. This fully Lagrangian advection algorithm introduces no artificial numerical dissipation or interpolation of parcel values. In the zero-viscosity limit, it preserves all of the Casimir invariants associated with parcel rearrangement. In the two-dimensional case presented here, these invariants are arbitrary piecewise continuous functions of the vorticity and concentration fields. The initial parcel centroids are evolved in a Lagrangian frame, using the method of characteristics. At any time this Lagrangian solution may be viewed by projecting it onto an Eulerian grid using a rearrangement map. The resulting rearrangement of initial parcel values is accomplished with a weighted Bresenham algorithm, which identifies quasi-optimal, distributed paths along which chains of parcels are pushed to fill in nearby empty cells. The error introduced by this rearrangement does not propagate to future time steps.