Flux limiter

Remarks

Take $u_t + a u_x = 0$ with $a > 0$ and $0 \le \nu \le 1$, with upwind and Lax–Wendroff fluxes. Then Sweby's condition $$0 \le \varphi(r) \le \min(2r, 2) \quad \text{for } r > 0$$ makes the scheme total variation diminishing. If moreover $\varphi(1) = 1$ and $\varphi$ is Lipschitz near $1$, the scheme is second-order accurate in smooth monotone regions. Example: the minmod limiter $\varphi(r) = \max(0, \min(1, r))$ vanishes for $r \le 0$, satisfies $\varphi(r) \le \min(r, 1) \le \min(2r, 2)$ for $r > 0$, and $\varphi(1) = 1$. Other standard limiters are superbee $\varphi(r) = \max(0, \min(2r, 1), \min(r, 2))$ and van Leer $\varphi(r) = (r + \abs{r}) / (1 + \abs{r})$. Non-example: the constant function $\varphi \equiv 1$ is not a flux limiter, because $\varphi(r) \ne 0$ for $r \le 0$; it gives back the unlimited Lax–Wendroff scheme, which is not TVD. At extrema $r_j \le 0$, so the scheme reduces to the low-order scheme there.