Summary: An efficient method for scalar advection is developed that selectively preserves monotonicity. Monotonicity preservation is applied only where the scalar field is likely to contain discontinuities as indicated by significant grid-cell-to-grid-cell variations in a smoothness measure conceptually similar to that used in weighted essentially non-oscillatory methods. In smooth regions, the numerical diffusion associated with monotonicity-preserving methods is avoided. The resulting method, while not globally monotonicity preserving, allows the full accuracy of the underlying advection scheme to be achieved in smooth regions. The violations of monotonicity that do occur are generally very small, as seen in the tests presented here. Strict positivity preservation may be effectively and efficiently obtained through an additional flux correction step.
The underlying advection scheme used to test this methodology is a variant of the piecewise parabolic method that may be applied to multi-dimensional problems using density-corrected dimensional splitting and permits stable semi-Lagrangian integrations using Courant-Friedrichs-Levy numbers larger than one. Two methods for monotonicity preservation are used here: flux correction and modification of the underlying polynomial reconstruction.