$5\mathrm{.}1$ In a steady two$-$dimensional situation, the variable $$ is governed by

div$(u)=$div$(grad)+a-b,$

$110$

Figure $5\mathrm{.}16$ Boundary conditio

Problem $5\mathrm{.}1.$

where $=1,=1,a=10,$ and $b=2.$ The flow field is such that $u=1$ and $v=4$

everywhere. For the uniform grid shown in Fig. $5\mathrm{.}16,x=y=1.$ The values of $$ are

given for the four boundaries. Adopting the control$-$volume design according to Practice

A in Section $4\mathrm{.}6-1,$ calculate the values of ${}_1,{}_2,{}_3,$ and ${}_4$ by use of:

$($a$)$ The central$-$difference scheme

$($b$)$ The upwind scheme

$($c$)$ The hybrid scheme

$($d$)$ The power$-$law scheme