Although first-order finite difference methods are monotonic and stable, they are also strongly dissipative, causing the solution to become smeared out. Second-order techniques are less dissipative, but are susceptible to nonlinear, numerical instabilities that cause nonphysical oscillations in regions of large gradient. The usual way to deal with these types of oscillation is to incorporate artificial diffusion into the numerical scheme. However, if this is applied uniformly over the problem domain, and enough is added to dampen spurious oscillations in regions of large gradient, then the solution is smeared out elsewhere. This difficulty is also touched upon in Section 12.3.1. The FCT technique is a scheme for applying artificial diffusion to the numerical solution of a convectively-dominated flow problem in a spatially nonuniform way. More artificial diffusion is applied in regions of large gradient, and less in smooth regions. The solution is propagated forward in time using a second-order scheme in which artificial diffusion is then added. In regions where the solution is smooth, some or all of this diffusion is subsequently removed, so the solution there is basically second order. Where the gradient is large, little or none of the diffusion is removed, so the solution in such regions is first order. In regions of intermediate gradient, the order of the solution depends on how much of the artificial diffusion is removed. In this way, the FCT technique prevents nonphysical extrema from being introduced into the solution.