On dynamic computational subgrid modeling

Abstract

In this paper we study a subgrid model based on extrapolation of a corrective force, in the case of a linear convection-diffusion problem $Lu=f$ in one dimension. The running average $u^{h}$ of the exact solution $u$ on the finest computational scale $h$ satisfies an equation $L_{h}u^{h}=[f]^{h}+F_{h}$, where $L_{h}$ is the operator used in the computation on the scale $h$, $[f]^{h}$ is the approximation of $f$ on the scale $h$, and $F_{h}$ acts as a corrective force, which needs to be modeled. The subgrid modeling problem is to compute approximations of $F_{h}$ without using finer scales than $h$. In this study we model $F_{h}$ by extrapolation from coarser scales than $h$ where the corrective force is directly computed with the finest scale $h$ as reference. We show in experiments that a corrected solution with subgrid model on scale $h$ corresponds to a non-corrected solution on less than $h/4$