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For incompressible flows a scheme was proposed for the convective facial values consisting of a linear combination of the upwind scheme and the facial values \overline{\phi }_f [36]:
Here, \gamma is a piecewise linear function of \widetilde{\phi }_P:
| \displaystyle \widetilde{\phi_P} \le 0: \hspace{1 cm} | \displaystyle \gamma=0 \; (Upwind Difference) | (578) |
| \displaystyle 0 < \widetilde{\phi_P} < \beta_m: \hspace{1 cm} | \displaystyle \gamma=\widetilde{\phi_P}/\beta_m | (579) |
| \displaystyle \beta_m \le \widetilde{\phi_P} < 1: \hspace{1 cm} | \displaystyle \gamma=1 \; (Central Difference) | (580) |
| \displaystyle 1 \le \widetilde{\phi_P}: \hspace{1 cm} | \displaystyle \gamma=0 \; (Upwind Difference) | (581) |
Due to the second term in the above equation \overrightarrow{\phi }_f is a nonlinear function of \phi_P. \beta_m should be in the range 0.1 \le \beta_m \le 0.5. In CalculiX \beta_m=0.1.
For the velocity \boldsymbol{v} the scalar \vert\vert\boldsymbol{v} \vert\vert is used to calculate \gamma. Since
| ||\boldsymbol{v}||,_j = \frac{v_i}{||\boldsymbol{v}|| } v_{i,j} | (582) |
one obtains
| \widetilde{||\boldsymbol{v}||}_P = 1 - \frac{(||\boldsymbol{v}||_E - ||\boldsymbol{v}||_P) ||\boldsymbol{v}||_P }{2 \boldsymbol{v}_P \cdot \nabla \boldsymbol{v}_P^T \cdot \boldsymbol{j}_\xi d(P,E) } . | (583) |