3.2.2 Flux-splitting

Both signs of the velocity are now possible.

Flux-splitting allows the use of schemes written for one sign of the velocity, still guaranteeing proper upwinding,

$$f \! \left( q \right) = f^{+} \! \left( q \right) + f^{-} \! \left( q \right)$$ (182)

with

$$\frac{\mathrm{d} f^{+}\!}{\mathrm{d} q} \ge 0 \enspace , \... ...ace \frac{\mathrm{d} f^{-}\!}{\mathrm{d} q} \le 0 \enspace .$$ (183)

E.g.: extension of FTBS scheme to Courant-Isaacson-Rees scheme (CIR, now stable for both signs of $v_{}$):

$$f_{i+\frac{1}{2}}^n = \left\{ \begin{array}{ll} f \! \le... ...mbox{if} \enspace v_{i+\frac{1}{2}} \le 0 \end{array} \right.$$ (184)

CIR for Burgers' equation:

$$f_{i+\frac{1}{2}}^n = \left\{ \begin{array}{ll} \frac{1}... ...mbox{if} \enspace v_{i+\frac{1}{2}} \le 0 \end{array} \right.$$ (185)