A simple but accurate mass conservative, peak-preserving, mixing ratio bounded advection algorithm with FORTRAN code

A simplified but very accurate method for calculating advection of mixing ratios in a mass conservative and monotonic manner is presented. This scheme replaces the polynomial approximations of other advection algorithms with dual-linear segments, and employs a special treatment near local maxima and minima to preserve extremes very well. Before updating mixing ratios, fluxes at grid-cell faces are bounded to yield conservative, monotonic, maximum-bounded and minimum-bounded solutions to the advection equation with very little numerical diffusion, even in locally deformational flows. “Bounded” here means that updated mixing ratios never exceed local maxima or fall below local minima. If an initial tracer distribution is everywhere positive, the solution is positive-definite, but negative values can be advected. With a “sharpening” option fully enabled, local maxima or minima are usually perfectly preserved, and features as small as 2–3Δx in width are advected with virtually no numerical diffusion for many Courant numbers and advection distances. Algorithms are presented for advecting mixing ratios through variable-spaced grids in any flow, including strongly deformational flows. One- and two-dimensional tests of the scheme are presented. Relative to other advection algorithms, this scheme yields significantly lower peak, distribution and rms errors, especially when advecting poorly resolved features with sizes 1–3Δx. Therefore this scheme may be useful in applications where peak and minima preservation of small features is desirable. A 41-line FORTRAN advection subroutine which can be readily applied for general advection problems is provided in the appendix.