3.2.11 Transformation of shock speed formula

Velocity at cell boundary from shock speed (Eq. (169) and Eq. (180), entropy fix ignored),

$$v_{i+\frac{1}{2}} = \frac{f \! \left( q_{i+1}^n \right) - f \! \left( q_i^n \right)} {q_{i+1}^n - q_i^n}$$ (190)

$$= \frac{f \! \left( q_{i+1}^n \right) - \frac{f \! \left( q_i^n \ri... ...t) + f \! \left( q_{i+1}^n \right)}{2}} {q_{i}^n - \frac{q_i^n + q_{i+1}^n}{2}}$$ (191)

$$= \frac{f \! \left( q_{{i_{\rm up}}_{i+\frac{1}{2}}}^n \right) - \f... ...\right)}{2}} {q_{{i_{\rm up}}_{i+\frac{1}{2}}}^n - \frac{q_i^n + q_{i+1}^n}{2}}$$ (192)

with the upwind index

$${i_{\rm up}}_{i+\frac{1}{2}} := \left\{ \begin{array}{ll... ...i+\frac{1}{2}} \le 0 \enspace \end{array} \right. \enspace .$$ (193)