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<br />22 <br /> <br />5. Numerical Scheme: <br /> <br />FLUVIAL-ll first solves the water-continuity equation, (2-26), and <br />momentum equation, (2-28), by an iterative, four-point, implicit, finite- <br />difference scheme developed by Arnein and Chu (1975). The flow information is <br />next used to compute the sediment-transport rate from either (2-29) or (2- <br />30). The sediment-continuity equation, (2-27), is then solved to <br />obtain AA in the following way: from (2-27) <br />c <br /> <br />AAc <br /> <br />At aQs <br />= -I-A (ax- qs) <br /> <br />....(2-31) <br /> <br />l(j j+l) <br />qs.= 2 qs. + qs. <br />1 1 1 <br />Qj + Qj+1 <br />IS. s. <br />[ 1 2 1 <br />Ax. 1 <br />1- <br /> <br />. ....(2-32) <br /> <br />aQs <br />(ax).= <br />1 <br /> <br />Qj + Qj+l <br />s. 1 s. 1 <br />1- 1-] <br />2 <br /> <br />....(2-33) <br /> <br />Qj + Qj+l _ Qj _ Qj+l <br />t S'l s'l S1' S1' <br />(AA ).=!L- [1- 1- + <br />c 1 I-A 2A x. 1 <br />1- <br /> <br />. '+1 <br />qJ + qJ <br />s. S. <br />1 1 <br />2 <br /> <br />] <br /> <br />.... (2-34) <br /> <br />Note that a backward-difference scheme was used in x and a forward-difference <br />scheme was used in t. The quantity AA obtained from (2-34) is then corrected <br />c <br /> <br />for the following effects: <br /> <br />(1) Adjustment in channel width: <br />Width adjustments are made in such a way that the spatial variation <br />in power expenditure per unit channel length (yQ5) is reduced along <br />the channel. The width is adjusted until the value which gives <br />minimum total stream power (integration of yQ5 over the reach length) <br />at each time step is found. To determi ne the wi dth change at each <br />section, the actual energy gradient at this section 5. is compared <br />1 <br />with the weighted, average energy gradient Siof its adjacent <br />sections given by <br /> <br />I <br />I <br />. I <br /> <br />S.= (5. lAX. + S'+IAX. 1 )/(2(AX. 1+ AX.)) <br />1 1- 1 1 1- 1- 1 <br />