I am a little confused about the connection between variables for the plain advection pde:$$u_t+au_x=0$$So initially I thought $x$ and $t$ are independent and $u$ is a function of those, but then we can write the same PDE and say that there are characteristics and then we have ODE for them:$$\frac{dx}{dt}=a$$So, now $x$ is a function of $t$. So are they dependent or not?
hyperbolic equation and characteristics
hyperbolic pde
You are correct in your initial assumption. x and t are independent and u is  the dependent variable. Your confusion is because you are assuming that $\frac{dx}{dt} = a$ is the solution to the equation. Notice that $u$ is not in this ODE. Basically this is an equation for the symmetry of the PDE. After getting this equation, you can write u = F(x-at)which is the actual solution since you will know u at t = 0 by specifying the initial conditions. x and t are not dependent on each other. $\frac{dx}{dt} = a$ merely states that the solution is fully specified on lines that are solutions of this ODE, and thus are called characteristics. 