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Consider a nonlinear hyperbolic conservation equation: $$ \partial_{t}u = -\partial_{x}f(u) $$ The spatial derivative of $f(u)$ may approximated after a spatial discretization by $x_{j}=j\Delta x$ $$ \partial_{x}f(u_{j}) \approx \frac{1}{\Delta x}\left(F_{j+\frac{1}{2}}-F_{j-\frac{1}{2}}\right) $$ where $F$ represents a numerical flux function that may depend on neighboring points and holds the same form from one cell to the next. In order to avoid Gibbs phenomena, the idea is to combine a low-order flux and high-order (2nd or greater) flux and combine them with a limiter function $\phi$ $$ F = F_{L} - \phi(r)(F_{H}-F_{L}) $$ where $$ r_{i} = \frac{u_{i+1}-u_{i}}{u_{i}-u_{i-1}} $$ The goal is preserve monotonicity by keeping the total variation (TV) of the solution non-increasing. I understand how the application of the TVD condition leads to the condition on the limiter: $$ \begin{cases} 0 \leq \phi(r_{i}) \leq 2r & r<1 \\ 0 \leq \phi(r_{i}) \leq 2 & 1 \leq r \end{cases} $$

But I do not understand at all the conditions required to keep the scheme 2nd-order and TVD. Could someone please show me how that region (see image) is derived?

enter image description here

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