A simple numerical method for solving the Navier$-$Stokes equations for constant prop$-$

erty flow advances the solution in small time steps $$t$,$ starting from the initial condition

U$($x$,0).$ On the nth step the numerical solution is denoted by Un$($x$),$ which approximates

U$($x$,$ n$$t$).$ Each time step consists of two sub$-$steps, the first of which yields an interme$-$

diate result $$Un$+1($x$)$ defined by:

$$U n$+1$

j $=$ U n

j $+$t

$($

$2$U n

j

$$xi$$xi

$$ U n

k

$$U n

j

$$xk

$)$

$(12)$

The second sub$-$step is:

U n$+1$

j $=$U n$+1$

j $$t$$n

$$xj

$(13)$

where n$($x$)$ is scalar field.

i$)$ Comment on the connection between $(12)$ and the Navier$-$Stokes equations:

DU

Dt $=1$

$$p $+$ $2$U

ii$)$ Assuming that Un is divergence$-$free, obtain from $(12)$ an expression $($in terms of

Un$)$ for the divergence of $$Un$+1.$

iii$)$ Obtain from $(13)$ an expression for the divergence of Un$+1.$

iv$)$ Hence, show that the requirement $$ Un$+1=0$ is satisfied if and only if n$($x$)$

satisfies the Poisson equation:

$2$n $=$U n

k

$$xj

$$U n

j

$$xk

v$)$ What is the connection between n$($x$)$ and the pressure?