Accuracy and Nonoscillatory Properties of Enslaved Difference Schemes

We describe a general method for modifying a given finite-difference scheme by representing the effects of the subgrid scales in terms of the resolved scales through enslaving. The new scheme is more accurate in certain parameter regimes while retaining the time-step stability of the original scheme. We consider two general enslaving relations: an approximate enslaving based on truncation analysis and an exact enslaving based on the dynamics of the governing equation. We find that the modified schemes based on the exact enslaving eliminate unphysical oscillations, producing monotone solutions even when the original difference schemes do not, have this property. We offer a truncation analysis to justify this property. We apply our enslaving technique to advection–diffusion equations in both one and two spatial dimensions.