$1.$Use the Taylor Series method$/$Taylor Table to derive a third order accurate scheme for a $1$st derivative.

Please use $4$ grid points in a partially upwind skewed stencil, i$.$e$.,$ two points to the left of the point of

interest, the point of interest itself, and one point to the right of the point of interest.

Be sure to verify that your derived scheme is third order accurate $($not second or fourth, e$.$g$.).$

$2.$ Find the effective wave number for this method. Plot the real and imaginary components of keff,

comparing with the exact wave number and comment on any differences.

$3.$ It is common, especially when deriving finite volume methods, to use points between grid points,

e$.$g$.$ ui$+1/2$ui$1/2$

dx and then defining the mid$-$points $(($i $+1/2)$ and $($i $1/2)$ in this example$)$ using

an interpolation method of choice. Use this approach and $3-$point upwind Lagrange interpolation on

a uniform grid to define the $(1)/(2)$grid points, then analyze this method to determine its Taylor Series

accuracy and discuss. Hint: the $3-$point upwind Lagrange interpolation will use $1$ grid point to the

right and $2$ to the left of the $(1)/(2)$grid point.