Enslaved Finite Difference Schemes for Nonlinear Dissipative PDEs

We show how the accuracy of a given finite difference scheme approximating a dissipative nonlinear PDE may be improved. The numerical solutions are decomposed into two parts that may be interpreted as approximating the large and small scales of the true solutions. By enslaving the small scales in terms of the larger ones, we derive a new difference scheme that is, in general, more accurate than the original scheme. The new scheme is also more computationally efficient, provided that the time derivatives of the problem are not too large.