ABSTRACT
Adding Eqs. (I 1.115a) and (11.115b) yields (fr + afx) + (gt +agJ = 0 (I 1.117)
which applies along dx/dt = +a, and adding Eqs. (11.116a) and (11.116b) yields (11.118)
which applies along dx/dt = -a. _ The spatial flux derivatives (i.e., fx and gJ in Eqs. (11.117) and (11.118) are
associated with positive-traveling waves and negative-traveling waves, respectively. Consequently, they should be differenced in the appropriate upwind directions. Let's attach superscripts + and - to the spatial flux derivatives in Eqs. (11.117) and (11.118), respectively, to remind us that they are associated with positive-traveling and negativetraveling waves, respectively. Thus,
cJ; + al/) + (gt + ag;) = 0 (fr - afx-) - (gt - ag;) = 0
(11.119) (11.120)
Solving Eqs. (11.119) and (11.120) explicitly for fr and gt yields the final form of the fluxvector-split PDEs:
(11.121)
(11.122)
Equations (I 1.121) and (11.122) are in a form suitable for developing upwind finite difference approximations. First-order FDEs can be developed using finite difference approximations such as Eq. (I 1.51), and second-order FDEs can be developed using finite difference approximations such as Eq. (11.55).
