Relation Between SGS Model and Numerical Discretization

Abstract

Considering a generic scalar nonlinear transport equation for the variable v

$$\frac {\partial v}{\partial t}+\frac {\partial F({v})}{\partial x}=0$$

the representation of the continuous solution v(x) by the grid function v _N results in a subgrid-scale error or residual

$${\mathcal{G}}_{SGS}=\frac {\partial {F}_N({v}_N)}{\partial x}-\frac {\partial {F}_N({v})}{\partial x}$$

which arises from the nonlinearity of F(v). The modified differential equation (MDE) for v _N is

$$\frac {\partial {v}_N}{\partial t}+\frac {\partial {F}_N({v_N})}{\partial x}={\mathcal{G}}_{SGS}.$$

Since for LES the ratio between characteristic flow scale and grid size h never can be considered as asymptotically small $\mathcal{G}_{SGS}$ cannot be neglected for proper evolution of v _N but requires approximation by modeling closures.

Once deconvolution operation and numerical flux function are determined, the modified-differential-equation analysis leads to an evolution equation of $\bar{u}_{N}$ in the form of

$$\frac {\partial \bar{u}_N}{\partial t}+G\ast \frac {\partial {F}_N({u}_N)}{\partial x}={\mathcal{G}}_{N},$$

where

$${\mathcal{G}}_N=G\ast \frac {\partial {F}_N({u}_N)}{\partial x}-G\ast \frac {\partial \tilde {F}_N({u}^\ast_N)}{\partial x}$$

is the truncation error of the discretization scheme. If ${\mathcal{G}}_{N}$ approximates $\bar{{\mathcal{G}}}_{SGS}$ in some sense we obtain an implicit subgrid-scale model implied by the discretization scheme.