Problem $1.$ For $(x),$ propose a third order finite difference $($FD$)$ discretization for $\frac{d}{dx}{|}_{x_i}$

using the values ${}_{i-2},{}_{i-1},{}_i,$ and ${}_{i+1}.$

Problem $2.$ For a function $(x,y),$ a consistent approximation for its second mixed deriva$-$

tive $\frac{{}^2}{\text{d}\text{e}\text{l}\text{x}{}_y}{|}_{x_i,y_i}$ in terms of the four points ${}_{i-1,j-1},{}_{i-1,j+1},{}_{i+1,j-1},$ and ${}_{i+1,j+1}$ is given as:

$\frac{de{l}^2}{\text{d}\text{e}\text{l}\text{x}\text{d}\text{e}\text{l}\text{y}}{|}_{x_i,y_i}=\frac{{}_{i-1,j-1}+{}_{i-1,j+1}+{}_{i+1,j-1}+{}_{i+1,j+1}}{4xy}+O((x{)}^2,(y{)}^2)$

If $=1,=-1$ and $=-1,$ find $.$

Problem $3.$ Check if the following scheme is consistent. If yes, find the order of accuracy.

$\frac{d^2}{d{x}^2}{|}_{x_i}=\frac{-{}_{i+3}+4{}_{i+2}-5{}_{i+1}+{}_i}{(x{)}^2}$

Problem $4.$

$($i$)$ In the common form of the transport equation discussed in this course, the term $gra{d}^2$

denotes the convection term. True or False.

$($ii$)$ What is meant by a well$-$posed problem? Discuss in $2-3$ lines.

$($iii$)$ One$-$D wave equation is which type of equation? Elliptic, parabolic or hyperbolic?

Problem $5.$ For the unsteady $1-$D heat conduction equation with left homogeneous Neu$-$

mann and right Dirichlet boundary conditions $($BCs$)$:

$($i$)$ Discretise it using the forward time centre space method. Specify appropriate discreti$-$

sations for the BCs too.

$($ii$)$ Specify the truncation error order. You do not need to derive it$,$ just specify it$.$

$($iii$)$ Perform the Von Neumann stability analysis for the above FTCS discretisation.

$($iv$)$ Write an algorithm$/$pseudocode of how you would implement the method.