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.
Best Answer
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