Meaning of imaginary eigenvalues in advection equation I'm reading the book Fluid Simulation for Computer Graphics (Amazon link), and stumped by the following sentence in Chapter 3: 

what's happening is that the eigenvalues of the Jacobian generated by the central difference are pure imaginary, thus always outside the region of stability.

Could someone explain what is meant here? What's the Jacobian matrix and why are the eigenvalues imaginary? What's the region of stability?
 A: Consider the one-dimensional flow of quantity $q(x,t)$. The advection equation is generalized to,
$$
\partial_tq+\partial_xF(q,x)=0\tag{1}
$$
where $F(q,x)$ is the flux of $q$ in the flow--in the case of the advection equation $F(q,x)=qu$. However, we can use the chain rule to write the above as
$$
\partial_tq+\frac{\partial F}{\partial q}\frac{\partial q}{\partial x}+\frac{\partial F}{\partial x}=0
$$
The $\partial_qF$ term is your Jacobian. If $q(x,t)$ is a set of values  then $\partial_qF$ is a matrix:
$$
J\equiv\frac{\partial F}{\partial q}=\left(\begin{array}{ccc}\frac{\partial F_{q_1}}{\partial q_1} & \frac{\partial F_{q_1}}{\partial q_2}&\cdots \\\frac{\partial F_{q_2}}{\partial q_1} & \frac{\partial F_{q_2}}{\partial q_2} &\cdots\\ \vdots&\vdots&\ddots
\end{array}\right)
$$
Since this is a matrix, you can find the eigenvalues which can either be real, imaginary, or zero. If the eigenvalues are imaginary, then you have and advection of the form
$$
\partial_tq+iA\partial_xq=0
$$
which does not really make physical sense: how does something flow with a velocity $iA$?
For stability analysis, I am not sure how exactly the Jacobian plays a role (maybe someone else can point it out?), I've only ever seen von Neumann stability analysis applied to the discretization. For example, consider a one dimensional, constant velocity flow; then Equation (1) is
$$
\partial_tq+u\partial_xq=0
$$
In terms of the central differences, this is
$$
\frac{q_i^{n+1}-q_i^n}{\Delta t}+\frac{u}{2}\frac{q_{i+1}^n-q_{i-1}^n}{\Delta x}=0\tag{2}
$$
where $\Delta t$ is your timestep and $\Delta x$ the width of your cells. If you solve for $q^{n+1}_i$ and assume that
$$
q(x_i,t_n)=G^nq_0e^{ikx_i}
$$
Then (2) becomes (prove this yourself!)
$$
q_i^{n+1}=\sqrt{1+\left(u\frac{\Delta t}{\Delta x}\sin[k\Delta x]\right)^2}\,q^n_i
$$
For stability, we require that the square-root term (called the amplification factor) be less than or equal to 1. In the above, no matter the choice of $\Delta t$ and $\Delta x$, it is always greater than 1, which means that the growth of $q$ is unconditionally unstable for this central scheme.
