A generalized numerical scheme for the linear hyperbolic equation $u_t + au_x = 0$ has the following form $$ \frac{u_j^{n+1} - u_j^n}{\Delta t} + a\frac{u_{j+1}^{n} - u_{j-1}^n}{2\Delta x} - \chi \frac{u_{j+1}^{n} - 2u_{j}^n + u_{j-1}^n}{\Delta x^2} = 0 . $$ a) Find the values of $\chi$ so that the above scheme is the central-explicit, forward, backward, Lax-Friedrich and Lax-Wendroff scheme.
b) What is the order of truncation for each scheme?
c) What is the stability condition for each of these schemes?
d) In what range of $\chi$ the scheme is unconditionally unstable?
e) Classify the schemes in (a) as (i) dissipative, and (ii) dispersive.
ATTEMPT
(a) Notice that if $\chi = 0$, then we obtain the central explicit scheme. If we write the numerical discretization as
$$u_j^{n+1} = u_j^n - \frac{ a \Delta t }{2 \Delta x} (u_{j+1}^n - u_{j-1}^n) + \frac{ \chi \Delta t }{\Delta x^2} (u_{j+1}^n - 2 u_j^n + u_{j-1}^n )$$
And one can see that if $\chi = a \Delta t/2$, then we have the LW scheme. If $\chi = a \Delta x/2$, then we have
$$ u_j^{n+1} = u_j^n + \frac{ a \Delta t }{\Delta x} (u_{j+1}^n - u_j^n) $$
which is forward explicit and if $\chi = - a \Delta x/2$, we obtain backward explicit. Now if $\chi = \frac{ \Delta x^2}{2 \Delta t }$, then we obtain Lax-Friedrichs.
We go to part (c) as (b) is trivial. Perhaps we can combine (c) and (d) in a single problem if we apply the discrete fourier transform:
$$ \hat{u}^{n+1} = \hat{u}^n - r( e^{ij \xi} -e^{-ij \xi } ) \hat{u}^n + \frac{ 2 \chi r }{a \Delta x} (e^{ij \xi} -e^{-ij \xi } -2 ) \hat{u}^n$$
Now, am I on the right track here? Is this the correct approach? As for e), I need some suggestions, how would we approach it?