I've been trying to read Advanced Numerical Approximation of Nonlinear Hyperbolic Equations (Ed. by A. Quarteroni, Springer, 1998, doi:10.1007/BFb0096351) to get a general idea of WENO for the scalar/system of conservation law.
I'm mainly trying to figure out how to calculate $u_{i + 1/2}^{\pm},u_{i - 1/2}^{\pm}$, which appears in the differential equation $$\frac{d\bar{u}}{dt}=\frac{1}{\Delta x}\bigg( \hat{f}_{i + 1/2}- \hat{f}_{i - 1/2}\bigg),$$ through the fact $$ \hat{f}_{i + 1/2} = h(u_{i + 1/2}^-,u_{i + 1/2}^+),\qquad \hat{f}_{i - 1/2} = h(u_{i - 1/2}^-,u_{i - 1/2}^+) ,$$ where $h$ is a numerical flux.
In his book he defines $u_{i + 1/2} = -\frac{1}{6}\bar{u}_{i-1}+ \frac{5}{6}\bar{u}_{i} + \frac{1}{3}\bar{u}_{i+1}$. From this I believe that $u_{i - 1/2} = -\frac{1}{6}\bar{u}_{i-2}+ \frac{5}{6}\bar{u}_{i-1} + \frac{1}{3}\bar{u}_{i}$. However now it is mysterious to me how I would get $u_{i + 1/2}^{\pm},u_{i - 1/2}^{\pm}$ from these formulas.
Any suggestions/help would be greatly appreciated.
In fact, this step can be the source of mistakes and confusion. Another good read is provided in (1). There, we learn that
Here, $I_i = (x_{i-1/2}, x_{i+1/2})$ represents a cell. Looking at section 2.2, we learn that the WENO reconstruction $u^-_{i+1/2}$ from stencils one point biased to the left reads $$ u^-_{i+\frac12} = w_1 u^{(1)}_{i+\frac12} + w_2 u^{(2)}_{i+\frac12} + w_3 u^{(3)}_{i+\frac12} . $$ The nonlinear weights $w_j > 0$ are given by the formula $$ w_j = \frac{\tilde w_j}{\sum_j \tilde w_j} ,\qquad \tilde w_j = \frac{\gamma_j}{(10^{-6} + \beta_j)^2} , $$ with the coefficients $$ \gamma_1 = \frac1{10}, \quad \gamma_2 = \frac3{5}, \quad \gamma_3 = \frac3{10}, $$ and smoothness indicators \begin{aligned} \beta_1 &= \frac{13}{12}(\bar u_{i-2} - 2\bar u_{i-1} + \bar u_i)^2 + \frac14(\bar u_{i-2} - 4\bar u_{i-1} + 3\bar u_{i})^2,\\ \beta_2 &= \frac{13}{12}(\bar u_{i-1} - 2\bar u_{i} + \bar u_{i+1})^2 + \frac14(\bar u_{i-1} - \bar u_{i+1})^2,\\ \beta_3 &= \frac{13}{12}(\bar u_{i} - 2\bar u_{i+1} + \bar u_{i+2})^2 + \frac14(3\bar u_{i} - 4\bar u_{i+1} + \bar u_{i+2})^2. \end{aligned} For each three-cell sub-stencil, the polynomial approximation of the cell interface value is deduced from \begin{aligned} u^{(1)}_{i+\frac12} &= \frac13 \bar u_{i-2} - \frac76 \bar u_{i-1} + \frac{11}6 \bar u_{i} ,\\ u^{(2)}_{i+\frac12} &= -\frac16 \bar u_{i-1} + \frac56 \bar u_{i} + \frac{1}3 \bar u_{i+1} ,\\ u^{(3)}_{i+\frac12} &= \frac13 \bar u_{i} + \frac56 \bar u_{i+1} - \frac{1}6 \bar u_{i+2} . \end{aligned} The formulas for $u^+_{i+1/2}$ can be obtained by taking the image of the above formulas for $u^-_{i+1/2}$ through a symmetry with respect to $x_{i+1/2}$, see (2, 3). At this stage it might help to draw a picture of the left-biased and right-biased stencils. The values of $u^\pm_{i-1/2}$ are obtained by substituting $i$ with $i-1$ in the above expressions.
(1) C.-W. Shu, "High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems", SIAM Rev. 51 (2009). doi:10.1137/070679065
(2) G.-S. Jiang, C.-W. Shu, "Efficient Implementation of Weighted ENO Schemes", J. Comput. Phys. 126 (1996). doi:10.1006/jcph.1996.0130
(3) C.-W. Shu, "Essentially Nonoscillatory and Weighted Essentially Nonoscillatory Schemes for Hyperbolic Conservation Laws", NASA/CR-97-206253, ICASE Report No. 97-65 (1997), or in: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations ed. by Cockburn et al. (1998), Springer. doi:10.1007/BFb0096351
See also:
C.-W. Shu, "Essentially non-oscillatory and weighted essentially non-oscillatory schemes", Acta Numerica (2020). doi:10.1017/S0962492920000057