A simplified but very accurate method for calculating advection of mixing ratios in a mass conservative and absolutely monotonic manner in divergent or nondivergent multidimensional flows is presented. This scheme uses a second-order-accurate, upstream approximation with monotone limiters and additionally adjusts fluxes at two cell edges around local extremes of a tracer distribution to significantly improve overall advection calculations. The minor flux adjustment slightly aggregates mass around local peaks in a manner which counters the inherent numerical diffusion associated with most numerical advection algorithms when advecting poorly resolved features. When advecting tracer shapes which are resolved by fewer than 10–20 grid cells, this scheme is significantly more accurate than higher-order algorithms for a wide range of test problems. For well-resolved tracer distributions this algorithm is very accurate and usually preserves local peak and minimum values almost perfectly. The scheme is positive-definite, but negative values can be advected with no modifications. A generalized algorithm and FORTRAN subroutine is presented for advecting mixing ratios or other conservative quantities through variable-spaced grids of one to three dimensions, including deformational flows. One- and two-dimensional tests are presented and compared with other higher-order algorithms. The computational requirements of this algorithm are significantly lower than those of other higher-order and less accurate schemes.