WENO interpolation ... [is] used ... to transfer information from one domain to another in a high order, nonoscillatory fashion
Define $h(x)$ implicitly via $$\frac{1}{\Delta x} \int_{x - \Delta x / 2}^{x + \Delta x / 2} h(s) \, ds = f(u(x))$$
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Putting the W in WENO: $$u^{{(\ast)}}_{{j + \frac{1}{2}}} = \frac{1}{10} u^{{(1)}}_{{j + \frac{1}{2}}} + \frac{6}{10} u^{{(2)}}_{{j + \frac{1}{2}}} + \frac{3}{10} u^{{(3)}}_{{j + \frac{1}{2}}}$$
What about the other letters?
What about shocks?
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Smoothness indicator when approximating a point in the interval $I = \left(x_j - \frac{\Delta x}{2}, x_j + \frac{\Delta x}{2}\right]$
Large $\beta$ indicates that the approximation $p(x)$ is not smooth nearby our point.
Recall the W in WENO: $$u^{{(\ast)}}_{{j + \frac{1}{2}}} = \frac{1}{10} u^{{(1)}}_{{j + \frac{1}{2}}} + \frac{6}{10} u^{{(2)}}_{{j + \frac{1}{2}}} + \frac{3}{10} u^{{(3)}}_{{j + \frac{1}{2}}}$$
Rather than using these directly, incorporate a penalty for each polynomial approximation's (lack of) smoothness: $\beta^{(1)}$, $\beta^{(2)}$, $\beta^{(3)}$.
TV-diminishing / strong stability preserving RK scheme
$$\begin{array}{c | c c c} 0 & & & \\ 1 & 1 & & \\ 1/2 & 1/4 & 1/4 & \\ \hline & 1/6 & 1/6 & 2/3 \end{array}$$