Consider a conservation law such as $u_t = -au_x$, and the following schematic. We can also write this as $u_t = -f_x,$ where $f=au$ is the flux.
Now, consider what happens when the wave moves.
But $f_w > f_e$, so we have more flux in than out, which is indicated by the red shaded regions. The result is an increase in the value of $u_P$. This is a problem, and it results in oscillations for the "standard" high order schemes. * Question: what about upwind? * Visualize the above schematic for upwind. What happens to cell P? Cell E? Consider in terms of the stepsize implied in the figure, and other step sizes.
"Linear numerical schemes for solving partial differential equations (PDE's), having the property of not generating new extrema (monotone scheme), can be at most first-order accurate." (From Wikipedia)
Flux limiters limit the flux to avoid this oscillation. For $u_t = f_x$, (and a simple time integration) we have
$$ u_i^{n+1} = u_i^n - \frac{\Delta t}{\Delta x}(f_e^n-f_w^n).$$We evaluate the flux at, e.g., the east face, as
$$f_e = f_e^{l} - \phi_e(r)(f_e^{l}-f_e^{h}) = (1-\phi_e(r))f_e^l + \phi(r)f_e^h.$$This is a combination of two fluxes. $f_e^l$ is a low-order flux, such as an upwind flux (first order), and $f_e^h$ is a high order flux, such as a central or average flux (second order).
If we use upwind for $f_e^l$ and a central (average) flux for $f_e^h$, we get
$$f_e = \underbrace{f_e^{(1)}}_{\mbox{upwind}} - \phi_e(r)\left(\underbrace{f_e^{(1)}}_{\mbox{upwind}} - \underbrace{\frac{f_e^{(1)}+f_e^{(2)}}{2}}_{\mbox{average}}\right),$$or,
$$f_e = f_e^{(1)} + \frac{1}{2}\phi_e(r_i)\left(f_e^{(2)} - f_e^{(1)}\right).$$Here, $f_e^{(1)}$ is the upwind flux and $f_e^{(2)}$ is the downwind flux.
$\phi(r)$ is a flux limiter function, which interpolates (nonlinearly) between the low and high fluxes.
The quantity $r$ is given by
The following graph shows regions where the function $\phi(r)$ can "live" so that the method will be "total variation diminishing" (TVD), and also be $O(\Delta x^2).$
Consider 4 limiter functions
The above figure uses a 3rd order RK method, with 200 grid points and $cfl = a\Delta t/\Delta x = 0.4.$