This work investigates the mitigation and elimination of scheme-related oscillations generated in compact and classical fourth-order finite difference solutions of stiff problems, represented here by the Burgers and Reynolds equations. The regions where severe gradients are anticipated are refined by the use of subdomains where the grid is distributed according to a geometric progression. It is observed that, for multi-domain solutions, both the classical and compact fourth-order finite difference schemes can exhibit spurious oscillations. When present, the oscillations are initially generated around the interface between the uniform and non-uniform grid subdomains. Based on a thorough study of the grid distribution effects, it is shown that the numerical oscillations are caused by inadequate geometric progression ratios within the non-uniformly discretized subdomains. Indeed, accurate solutions are obtainable if and only if the grid ratios in the non-uniform subdomains are greater than a critical threshold ratio. It is concluded that high-order classical and compact schemes can be used with confidence to efficiently solve one-or two-dimensional problems whose solutions exhibit sharp gradients in very thin regions, provided that the numerically generated oscillations are eliminated by an appropriate choice of grid distribution within the non-uniformly discretized subdomains.