You are required to write a computer program and solve the following $2$D steady$-$state diffusion

equation using the finite difference method:

$$

$2$

$$x

$2+$

$$

$2$

$$y

$2=$ S$$

on a square domain of unit length. The boundary conditions are as follows:

$(0,$ y$)=500$ exp$(50[1+$ y

$2$

$])$

$(1,$ y$)=100(1$ y$)+500$ exp$(50$y

$2$

$)$

$($x$,0)=100$x $+500$ exp$(50[1$ x$]$

$2$

$)$

$($x$,1)=500$ exp$(50\{[1$ x$]$

$2+1\})$

The source term is given by:

S$=50000$ exp$(50\{[1$ x$]$

$2+$ y

$2$

$\})(100\{[1$ x$]$

$2+$ y

$2$

$\}2)$

For reference, the analytical solution is given below:

$($x$,$ y$)=500$ exp$(50\{[1$ x$]$

$2+$ y

$2$

$\})+100$x$(1$ y$)$

$(1)$ Solve the $2$D steady$-$state diffusion equation using Gaussian elimination method for the

following grids: $21,41,$ and $81$ grid points in each direction. Show the computed $$ field as

a contour plot for the finest grid. Plot the CPU run time vs total number of grid points and

discuss the trend and comment on the computational efficiency of the Gauss elimination

method.

$(2)$ Solve the same $2$D steady$-$state diffusion equation using the Gauss$-$Seidel iterative method

for the following three grids: $41,81,$ and $161$ grid points in each direction. Show the

computed $$ field as a contour plot for the finest grid. Plot the residual vs number of

iterations for the three grids in the same plot. Compare and discuss the dependence of the

convergence rate on the total number of grid points. Plot the variation of CPU run time

with total number of grid points for the Gauss$-$Seidel iterative method and discuss the trend.

How do the CPU run times of the Gauss$-$Seidel iterative method compare with those of

Gauss elimination method?

$(3)$ Solve the same $2$D steady$-$state diffusion equation using the line$-$by$-$line method $($row

sweep$)$ for the following three grids: $41,81,$ and $161$ grid points in each direction. Please

note that you will need to use the TDMA solver to solve the resulting system of linear

equations because of the tridiagonal structure of the resulting coefficient matrices. Show

the computed $$ field as a contour plot for the finest grid. Plot residual vs number of

iterations for the line$-$by$-$line method and for the Gauss$-$Seidel method for the $161161$

grid $($both in the same plot$).$ Compare and discuss the trends. Plot the variation of CPU run

time with total number of grid points for the line$-$by$-$line method. How does the CPU run

times and the obtained scaling compare with those obtained for the Gauss elimination

method and point$-$wise iterative method?

$(4)$ Solve the same $2$D steady$-$state diffusion equation using the Alternating Direction Implicit

$($ADI$)$ method for the $4181$ grid and plot the residual vs number of iterations for row$-$wise

sweep, column$-$wise sweep, and ADI method in the same plot. Discuss the trends.

Solve in MATLAB