For the non-constant-coefficient advection-diffusion equation with decay a(∂tc + λc + u∂sc) - ∂s(aκ∂sc) = q, the non-uniformity can almost be removed with a local change of variables from c(s,t) to b(x,t). So, in numerical computations non-uniformity need not imply any reduction from optimal accuracy. A computational grid that was uniformly spaced with respect to s becomes non-uniform with respect to the intrinsic x-coordinate. Suitable non-uniform grid, optimal and near-optimal compact finite-difference schemes were derived in part II of this sequence of papers. The cylindrical advection-diffusion equation is used as an exactly solvable test case.