ppm_reconstruction

Family: finite_volume
Grid family: cartesian
Kind: scheme
Accuracy: O(dx³)
Applies to: reconstruct(q), dim=x
Rule file: discretizations/finite_volume/ppm_reconstruction.json
Tags: #finite-volume #ppm #reconstruction #colella-woodward

Stencil

Coefficients

Edge value at q_{i+1/2}:

selector	offset	coeff
`cartesian`	−1	`−1/12`
`cartesian`	0	`+7/12`
`cartesian`	+1	`+7/12`
`cartesian`	+2	`−1/12`

Per-cell parabola (CW84 eqs. 1.5, 1.7, 1.10):

a_L = q_{i−1/2} (right limit of cell i−1)
a_R = q_{i+1/2} (left limit of cell i+1)
da = a_R − a_L
a₆ = 6·(q_i − ½(a_L + a_R))
a(ξ) = a_L + ξ·(da + a₆·(1 − ξ)), ξ ∈ [0, 1]

Limiting (CW84 eqs. 1.10) is not applied at this rule level — see the flux-limiter rules (flux_limiter_minmod, flux_limiter_superbee).

Discrete operator

The rule reconstructs a piecewise-parabolic profile inside each cell from cell-averaged inputs $\bar q_i = \tfrac{1}{\Delta x}\!\int_{x_i-\Delta x/2}^{x_i+\Delta x/2}\!\!q(x)\,dx$ on a uniform Cartesian axis. Reconstruction is two passes — edge values first, parabola coefficients second.

Edge-value pass (CW84 eq. 1.6). A 4-point centered combination of cell averages produces the 4th-order interpolant at $x_{i+1/2}$:

$$q_{i+1/2} \;=\; \frac{-\,\bar q_{i-1} \;+\; 7\,\bar q_i \;+\; 7\,\bar q_{i+1} \;-\; \bar q_{i+2}}{12}.$$

Symmetric Taylor expansion of the four neighbors about $x_{i+1/2}$ cancels the linear, quadratic, and cubic terms; the leading error is $-\,\tfrac{\Delta x^{4}}{60}\,q^{(4)}(x_{i+1/2})$, so the edge interpolant itself is fourth-order accurate in the cell average.

Parabola pass (CW84 eqs. 1.5, 1.7). Each cell is represented in a local coordinate $\xi = (x - x_{i-1/2})/\Delta x \in [0,1]$ by

$$a(\xi) \;=\; a_L \;+\; \xi\,\bigl(\Delta a \;+\; a_6\,(1-\xi)\bigr),$$

where $a_L = q_{i-1/2}$ and $a_R = q_{i+1/2}$ are the edge values from the first pass and

$$\Delta a \;=\; a_R - a_L, \qquad a_6 \;=\; 6\!\left(\bar q_i - \tfrac{1}{2}(a_L + a_R)\right).$$

The choice of $a_6$ is the cell-mean–conserving constraint $\tfrac{1}{\Delta x}\!\int a(\xi)\,d\xi = \bar q_i$ — the parabola exactly recovers the cell average regardless of the edge values.

Composing the two passes, point-evaluation at any subcell $\xi$ recovers $q(x_{i-1/2} + \xi\,\Delta x)$ with leading error $O(\Delta x^{3})$ in $L^\infty$ for smooth profiles. Limiting and discontinuity-detection (CW84 eqs. 1.10, 1.14–1.17) are deliberately omitted at this rule level so the unlimited fourth-order/parabola pair can be exercised in isolation; the flux-limiter rules above compose on top.

Convergence

The fixture under discretizations/finite_volume/ppm_reconstruction/fixtures/convergence/ sets expected_min_order = 2.8 to tolerate pre-asymptotic drift on the 16 → 32 → 64 → 128 sequence; the per-cell sub-sampling settles onto the third-order asymptote by n = 32.