Exercise Problems $4$: Finite$-$Difference Method $($Stability$,$ and Convergence$)$

Perform a von Neumann stability analysis and analyze convergence of BTCS finite different approximation of pure advection equation

$-\frac{C_r}{2}{f}_{j-1}^{n+1}+f_j^{n+1}+\frac{C_r}{2}{f}_{j+1}^{n+1}=f_j^n$

Consider the one$-$dimensional diffusion equation:

$\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{t}}=\frac{de{l}^{2}f}{del{x}^2}$

Perform a von Neumann stability analysis of the following finite difference approximation of this equation

$\frac{f_j^{n+1}-f_j^{n-1}}{2t}=\frac{f_{j+1}^n-2f_j^n+f_{j-1}^n}{x^2}$

The DuFont and Fraqnkel method for diffusion equation

$\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{t}}=\frac{de{l}^{2}f}{del{x}^2}$

is given as

$\frac{f_j^{n+1}-f_j^{n-1}}{2t}=\frac{f_{j+1}^n-(f_j^{n+1}+f_j^{n-1})+f_{j-1}^n}{x^2}$

or

$(1+2d)f_j^{n+1}=(1-2d)f_j^{n-1}+2d(f_{j+1}^n+f_{j-1}^n)$

where $d=\frac{t}{}{x}^2$ is the diffusion number. Perform the von Neumann stability analysis.

$4.$ Suppose it is possible to approximate $\frac{de{l}^{2}h}{del{x}^2}$ by unknown linear combination of $h_{j-2}^n,h_{j-1}^n,h_j^n,h_{j+1}^n,$ and $h_{j+2}^n$ :

$\frac{de{l}^{2}h}{del{x}^2}$~~$l{h}_{j-2}^n+m{h}_{j-1}^n+n{h}_j^n+p{h}_{j+1}^n+q{h}_{j+2}^n$

$($a$)$ Determine $l,m,n,p,$ and $q.$

$($b$)$ What is the order of truncation error?