Problem $1$

Elliptical partial differential equations $($PDEs$)$ are very important and common in many domains. For example,

ideal flows in fluid dynamics, where the governing equation for continuity can be transformed into a stream

function.

For the potential flow shown in the picture, the governing equation for the stream function $$ is:

$-=(grad*grad)-=gra{d}^2=(\frac{de{l}^2}{del{x}^2}+\frac{de{l}^2}{del{y}^2})=\frac{de{l}^2}{del{x}^2}+\frac{de{l}^2}{del{y}^2}=0$

at point $16$ and $15,=10,$ the $L$ shape on the top has $=0,$ and the other one at the bottom is $=20.$

Assume the grid spaces $x=y=2.$

A$).$ Write out the central finite difference $($FD$)$ discretization for Points $12,13,14,17,18,19.$

B$).$ By using the Taylor expansion series, prove the above equation is compatible$/$consistent with the original

PDE, i$.$e$.$ Eqn$\mathrm{.}1,$ if the mesh is very dense, i$.$e$.,x$ and $y$ are very small,

C$).$ Write the implicit $66$ matrix for the inner nodes $($no need to include boundary conditions$).-$ Then this

group of linear algebraic equations can be solved very conveniently with MATLAB.

D$).$ Giving $=0$ to the $6$ inner nodes as their initial values, use an explicit scheme to iterate the inner values

for the $6$ points $(4$ rounds of iterations for those nodes shall be enough$).$ You shall appreicate the boundary

effect propagating into the inner domain.