High resolution schemes

Figure: Element participation for High Resolution Schemes

To remedy the problem of spurious viscosity several high resolution schemes have been invented [61]. They nearly all consist of piecewise functions of the form

\widetilde{\overrightarrow{\phi }}_e = a \widetilde{\phi}_P + b, (565)

where a and b are constants and \widetilde{\phi_P } and \widetilde{\overrightarrow{\phi_e } } are defined as (cf. Figure , in which a regular mesh is shown):

\widetilde{\phi_P }:= \frac{\phi_P - \phi_W}{\phi_E-\phi_W} (566)

and

\widetilde{\overrightarrow{\phi_e } } := \frac{\overrightarrow{\phi_e}-\phi_W }{\phi_E-\phi_W}. (567)

Substituting Equations () and () in Equation () yields:

\overrightarrow{\phi_e} = a \phi_P + b \phi_E + (1-a-b) \phi_W. (568)

Figure: Setting for high resolution scheme in irregular mesh

For irregular meshes element centers W, P and E may not be aligned Therefore, usually a fictituous aligned position W' is assumed such that (Figure ):

\phi_{W'} = \phi_E - 2 \nabla \phi_P \cdot \boldsymbol{j}_\xi d(P,E) (569)

and used in Equation () instead of \phi_W leading to

\overrightarrow{\phi_e}= a \phi_P + (1-a) \phi_E - 2 (1-a-b) \nabla \phi_P\cdot \boldsymbol{j}_\xi d(P,E). (570)

The coefficients a and b in Equation () usually depend in a discrete way on the value of \widetilde{\phi }_P, which can be approximated by:

\widetilde{\phi }_P = 1 - \frac{\phi_E-\phi_P}{\phi_E-\phi_W} \approx 1 - \frac{\phi_E-\phi_P}{2 \nabla \phi_P \cdot \boldsymbol{j}_\xi d(P,E) }. (571)

For the Modified Smart Scheme, which is implemented in CalculiX, the following relationships are defined:

\displaystyle \widetilde{\phi_P} < 0: \hspace{1 cm} \displaystyle \widetilde{\overrightarrow{\phi } }_f = \widetilde{\phi }_P \; (a=1,b=0) (572)

\displaystyle 0 \le \widetilde{\phi_P} < 1/6: \hspace{1 cm} \displaystyle \widetilde{\overrightarrow{\phi } }_f = 3 \widetilde{\phi }_P \;(a=3,b=0) (573)

\displaystyle 1/6 \le \widetilde{\phi_P} < 7/10: \hspace{1 cm} \displaystyle \widetilde{\overrightarrow{\phi } }_f = 3/4 \widetilde{\phi }_P + 3/8 \;(a=3/4,b=3/8) (574)

\displaystyle 7/10 \le \widetilde{\phi_P} < 1: \hspace{1 cm} \displaystyle \widetilde{\overrightarrow{\phi } }_f = 1/3 \widetilde{\phi }_P + 2/3 \;(a=1/3,b=2/3) (575)

\displaystyle 1 \le \widetilde{\phi_P}: \hspace{1 cm} \displaystyle \widetilde{\overrightarrow{\phi } }_f = \widetilde{\phi }_P \; (a=1,b=0) (576)

Notice that by using the resulting equations () and () the convective interpolation at face e only depends on element center values in the neighboring elements P and E. Therefore, these formulas can be used in a completely irregular mesh. Finally, it is worth noting that if \phi is a vector field (e.g. the velocity) the above formulas are applied componentwise.