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Finite-volume MUSCL, nodal ENO2 and WENO5 reconstruction conventions

Definition

Let $h > 0$ , let $I_{i} = [i h, (i + 1) h)$ , let $x_{i} = (i + \frac{1}{2}) h$ be its center, and write $x_{i + 1 / 2} = (i + 1) h$ and $x_{i - 1 / 2} = i h$ for its right and left faces. A cell average of a locally integrable scalar function $u$ is $U_{i} = h^{- 1} \int_{I_{i}} u (x) d x$ . The cells are half-open only to form a partition; the choice of endpoint representatives does not change an integral.

Here MUSCL/minmod means the piecewise-linear reconstruction with the limiter defined next; nodal ENO2 means the two-stencil quadratic selector defined below; and cell-average WENO5-JS/WENO-Z means the three-candidate reconstructions with the normalized nonlinear weights displayed below. These names specify the constructions and do not, by themselves, assert an accuracy or stability property.

For real $a, b$ , define $minmod (a, b) = {\begin{cases} sgn (a) \min (∣ a ∣, ∣ b ∣), & a b > 0, \\ 0, & a b \leq 0. \end{cases}$ The MUSCL/minmod reconstruction assigns the cell slope $s_{i} = h^{- 1} minmod (U_{i} - U_{i - 1}, U_{i + 1} - U_{i})$ and the two face states $U_{i - 1 / 2}^{+, M} = U_{i} - h s_{i} / 2$ and $U_{i + 1 / 2}^{-, M} = U_{i} + h s_{i} / 2$ . Here $+$ denotes a trace from the right cell and $-$ a trace from the left cell.

For separate nodal data $V_{j} = u (y_{j})$ at $y_{j} = j h$ , define the second divided difference by $[y_{a}, y_{b}, y_{c}] V = (((V_{c} - V_{b}) / (y_{c} - y_{b})) - ((V_{b} - V_{a}) / (y_{b} - y_{a}))) / (y_{c} - y_{a})$ . At the face $y_{i + 1 / 2} = (i + \frac{1}{2}) h$ , set $d_{L} = ∣ [y_{i - 1}, y_{i}, y_{i + 1}] V ∣$ and $d_{R} = ∣ [y_{i}, y_{i + 1}, y_{i + 2}] V ∣$ . Choose the left triple when $d_{L} \leq d_{R}$ and otherwise choose the right triple. For the selected triple $(y_{a}, y_{b}, y_{c})$ define its Newton quadratic by $P_{a, b, c} (y) = V_{a} + \frac{V_{b} - V_{a}}{y_{b} - y_{a}} (y - y_{a}) + [y_{a}, y_{b}, y_{c}] V (y - y_{a}) (y - y_{b}) .$ The nodal ENO2 face value is $P_{a, b, c} (y_{i + 1 / 2})$ . This nodal interpolation branch is distinct from the cell-average WENO branch below.

For cell averages $U_{i - 2}, \dots, U_{i + 2}$ , define the left-biased WENO5 candidate face functionals $p_{0} = (2 U_{i - 2} - 7 U_{i - 1} + 11 U_{i}) / 6, p_{1} = (- U_{i - 1} + 5 U_{i} + 2 U_{i + 1}) / 6, p_{2} = (2 U_{i} + 5 U_{i + 1} - U_{i + 2}) / 6.$ Define the Jiang–Shu indicators $β_{0} = \frac{13}{12} (U_{i - 2} - 2 U_{i - 1} + U_{i})^{2} + \frac{1}{4} (U_{i - 2} - 4 U_{i - 1} + 3 U_{i})^{2},$ $β_{1} = \frac{13}{12} (U_{i - 1} - 2 U_{i} + U_{i + 1})^{2} + \frac{1}{4} (U_{i - 1} - U_{i + 1})^{2}, β_{2} = \frac{13}{12} (U_{i} - 2 U_{i + 1} + U_{i + 2})^{2} + \frac{1}{4} (3 U_{i} - 4 U_{i + 1} + U_{i + 2})^{2} .$ Fix a positive parameter $ϵ_{0}$ with the same units as each $β_{k}$ , independently of $h$ , and ideal weights $(d_{0}, d_{1}, d_{2}) = (1 / 10, 6 / 10, 3 / 10)$ . If $u$ is nondimensionalized, $ϵ_{0}$ is a fixed positive number. The WENO-JS weights and left face value are $α_{k}^{J} = d_{k} / (ϵ_{0} + β_{k})^{2}$ , $ω_{k}^{J} = α_{k}^{J} / \sum_{l = 0}^{2} α_{l}^{J}$ , and $U_{i + 1 / 2}^{-, J} = \sum_{k = 0}^{2} ω_{k}^{J} p_{k}$ . For the WENO-Z comparison, use the fifth-order $p = 2$ weights: with $τ_{5} = ∣ β_{0} - β_{2} ∣$ , set $α_{k}^{Z} = d_{k} [1 + (τ_{5} / (ϵ_{0} + β_{k}))^{2}]$ , $ω_{k}^{Z} = α_{k}^{Z} / \sum_{l = 0}^{2} α_{l}^{Z}$ , and $U_{i + 1 / 2}^{-, Z} = \sum_{k = 0}^{2} ω_{k}^{Z} p_{k}$ . All denominators are positive because $ϵ_{0} > 0$ and every $β_{k}$ is a sum of squares. The right face state is obtained by reflection about that face: replace each stencil value $U_{k}$ in the displayed left formula by $U_{2 i + 1 - k}$ .

For scalar linear advection $u_{t} + a u_{x} = 0$ , a direct reconstructed-upwind forward-Euler update uses $ν = a Δ t / h$ , the flux $F_{i + 1 / 2} = a U_{i + 1 / 2}^{-, M}$ when $a \geq 0$ and $F_{i + 1 / 2} = a U_{i + 1 / 2}^{+, M}$ when $a < 0$ , and $U_{i}^{n + 1} = U_{i}^{n} - (Δ t / h) (F_{i + 1 / 2} - F_{i - 1 / 2})$ . The total variation of a sequence is $TV (U) = \sum_{i \in Z} ∣ U_{i + 1} - U_{i} ∣$ when the sum is finite; TVD means that this quantity does not increase in one update.

For a vector system whose flux Jacobian at an interface is diagonalizable as $A = R Λ R^{- 1}$ with real diagonal $Λ$ , a characteristic-wise reconstruction applies a chosen scalar reconstruction to the components of $R^{- 1} U$ and maps the reconstructed vector back by $R$ . This definition alone gives no TVD or stability result for systems.

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Sources

Chi-Wang Shu, Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws, complete lecture notes (standard reference, not scraped)
Guang-Shan Jiang and Chi-Wang Shu, Efficient Implementation of Weighted ENO Schemes (standard reference, not scraped)
Marcos Castro, Bruno Costa and Wai Sun Don, High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws (standard reference, not scraped)
Jérôme Droniou, A Beginner’s Course in Finite Volume Approximation of Scalar Conservation Laws, complete lecture notes (standard reference, not scraped)