High resolution schemes

Figure 170: Element participation for High Resolution Schemes
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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

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

where $ a$ and $ b$ are constants and $ \widetilde{\phi_P }$ and $ \widetilde{\overrightarrow{\phi_e } }$ are defined as (cf. Figure 170, in which a regular mesh is shown):

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

and

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

Substituting Equations (566) and (567) in Equation (565) yields:

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

Figure 171: Setting for high resolution scheme in irregular mesh
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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 171):

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

and used in Equation (568) instead of $ \phi_W$ leading to

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

The coefficients a and b in Equation (565) usually depend in a discrete way on the value of $ \widetilde{\phi }_P$, which can be approximated by:

$\displaystyle \widetilde{\phi }_P = 1 - \frac{\phi_E-\phi_P}{\phi_E-\phi_W} \approx 1 - \frac{\phi_E-\phi_P}{2 \nabla \phi_P \cdot \boldsymbol{j}_\xi d(P,E) }.$ (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 (570) and (571) 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.