Fractal Model for Coarse-Grained Nonlinear Partial Differential Equations

Spatially coarse-grained (or effective) versions of nonlinear partial differential equations must be closed with a model for the unresolved small scales. For systems that are known to display fractal scaling, we propose a model based on synthetically generating a scale-invariant field at small scales using fractal interpolation, and then analytically evaluating its effects on the large, resolved scales. The procedure is illustrated for the forced Burgers equation, solved numerically on a coarse grid. Detailed comparisons with direct simulation of the full Burgers equation and with an effective viscosity model are presented.