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 twodimensional 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.