embedded3 item 10: the apertured cut-cell wall (AM-wall) — cutwall.rs classification, apertured projection with the compatible wall flux, cut predictor (V_u = αhA, averaged mass fluxes, implicit wall shear, inertia floor), cut load route; sphere MMS CutCell ≤ GhostBinary at n 12/24 (ratio 0.96), loads 9.3/10.7 % at n 24

Co-Authored-By: Claude Fable 5.1 <[email protected]>

crates/specialized/rtx-cfd/src/solvers/incompressible/embedded3/cutwall.rs

@@ -0,0 +1,400 @@
+//! The apertured cut-cell wall (`WallScheme::CutCell`, item 10 — the
+//! "AM-wall"): the cut geometry classifies the grid (a cell is fluid where
+//! its fluid volume is positive; an interior face is an unknown where its
+//! aperture is positive, prescribed the surface velocity otherwise — no
+//! ghost faces), and the wall enters the operators through the apertures:
+//! continuity `Σ_f A_f u_f·n_f + U_b·W_c = 0` per cell, the projection
+//! coefficient `dt·A_f/δ`, the momentum control volume of an unknown face
+//! `V_u = A_f h` with its own faces' apertures averaged from the two
+//! adjacent cells (mass fluxes averaged, so the momentum volume conserves
+//! mass exactly when the cells do), the wall closing it (`W = −Σ A n`),
+//! an implicit wall shear `μ A_w (u_f − U_b)/d_f`, and an inertia floor of
+//! 0.1 in the time derivative only. Every prescribed value is the limit of
+//! the computed one as the aperture closes (the shear coefficient grows as
+//! `1/A_f`), which is what makes the wall smooth in the interface position.
+
+use super::Grid;
+use super::body::Body;
+use super::cut::CutGeometry;
+use super::field::Field;
+use super::step::{Boundaries, Side};
+use super::wall::{FaceKind, Mask};
+
+/// The inertia floor: the momentum volume's fraction in the time
+/// derivative is at least this.
+pub(super) const INERTIA_FLOOR: f64 = 0.1;
+/// The wall-distance floor of a face, in units of the smallest spacing.
+pub(super) const DISTANCE_FLOOR: f64 = 0.05;
+
+/// Lattice addressing of faces and cells with the periodic wrap in z as
+/// data: a face of component `c` at `p = [i, j, k]` (its own coordinate is
+/// the face index, the others the cell's), a cell at `[i, j, k]`.
+#[derive(Debug, Clone, Copy)]
+pub(super) struct Lattice {
+    pub(super) g: Grid,
+    pub(super) periodic_z: bool,
+}
+
+impl Lattice {
+    fn wrap_z(&self, k: i64, planes: i64) -> Option<usize> {
+        if self.periodic_z {
+            Some(k.rem_euclid(self.g.nz as i64) as usize)
+        } else if (0..planes).contains(&k) {
+            Some(k as usize)
+        } else {
+            None
+        }
+    }
+
+    /// The index of the face of component `c` at `p`, `None` outside.
+    pub(super) fn face(&self, c: usize, p: [i64; 3]) -> Option<usize> {
+        let (nx, ny, nz) = (self.g.nx as i64, self.g.ny as i64, self.g.nz as i64);
+        let (i, j) = (p[0], p[1]);
+        let (ni, nj, nk) = match c {
+            0 => (nx + 1, ny, nz),
+            1 => (nx, ny + 1, nz),
+            _ => (nx, ny, nz + 1),
+        };
+        if !(0..ni).contains(&i) || !(0..nj).contains(&j) {
+            return None;
+        }
+        let k = self.wrap_z(p[2], nk)?;
+        let (i, j) = (i as usize, j as usize);
+        Some(match c {
+            0 => self.g.uface(k, j, i),
+            1 => self.g.vface(k, j, i),
+            _ => self.g.wface(k, j, i),
+        })
+    }
+
+    /// The index of the cell at `p`, `None` outside.
+    pub(super) fn cell(&self, p: [i64; 3]) -> Option<usize> {
+        let (nx, ny, nz) = (self.g.nx as i64, self.g.ny as i64, self.g.nz as i64);
+        if !(0..nx).contains(&p[0]) || !(0..ny).contains(&p[1]) {
+            return None;
+        }
+        let k = self.wrap_z(p[2], nz)?;
+        Some(self.g.cell(k, p[1] as usize, p[0] as usize))
+    }
+
+    /// The centre of the face of component `c` at `p`.
+    pub(super) fn face_position(&self, c: usize, p: [i64; 3]) -> [f64; 3] {
+        let h = [self.g.dx, self.g.dy, self.g.dz];
+        let mut x = [0.0; 3];
+        for d in 0..3 {
+            let off = if d == c { 0.0 } else { 0.5 };
+            x[d] = (p[d] as f64 + off) * h[d];
+        }
+        x
+    }
+}
+
+/// The geometry of an unknown face's momentum control volume.
+#[derive(Debug, Clone, Copy)]
+pub(super) struct CvGeometry {
+    /// The face's own aperture.
+    pub(super) alpha: f64,
+    /// The control volume's face apertures `[direction][minus, plus]`.
+    pub(super) ap: [[f64; 2]; 3],
+    /// The wall's vector area closing the control volume (into the body).
+    pub(super) wall: [f64; 3],
+    /// The wall distance of the face (floored).
+    pub(super) distance: f64,
+}
+
+impl Mask {
+    /// Classify the grid against `body` at `t` by its cut geometry.
+    pub fn build_cut(body: &Body, g: Grid, t: f64, b: Boundaries) -> Result<Self, String> {
+        let cut = CutGeometry::build(body, g, t);
+        let (nx, ny, nz) = (g.nx, g.ny, g.nz);
+        let periodic = b.z0 == Side::Periodic;
+        let allowed = |side: Side| matches!(side, Side::Velocity | Side::Periodic | Side::SlipWall);
+        let mut cell_fluid = vec![true; g.cells()];
+        let mut fluid_cells = 0;
+        let mut anchor = None;
+        for k in 0..nz {
+            for j in 0..ny {
+                for i in 0..nx {
+                    let idx = g.cell(k, j, i);
+                    let fluid = cut.vol[idx] > 0.0;
+                    cell_fluid[idx] = fluid;
+                    if fluid {
+                        fluid_cells += 1;
+                        if anchor.is_none() {
+                            anchor = Some(idx);
+                        }
+                    } else {
+                        let touches = (i == 0 && !allowed(b.x0))
+                            || (i + 1 == nx && !allowed(b.x1))
+                            || (j == 0 && !allowed(b.y0))
+                            || (j + 1 == ny && !allowed(b.y1))
+                            || (k == 0 && !allowed(b.z0))
+                            || (k + 1 == nz && !allowed(b.z1));
+                        if touches {
+                            return Err(format!(
+                                "embedded body reaches a domain side that is not a Velocity/Periodic side at cell ({k}, {j}, {i})"
+                            ));
+                        }
+                    }
+                }
+            }
+        }
+        let Some(anchor) = anchor else {
+            return Err("embedded body covers the whole domain".into());
+        };
+        let kind = |a: f64| {
+            if a > 0.0 {
+                FaceKind::Fluid
+            } else {
+                FaceKind::Solid
+            }
+        };
+        let mut u_kind = vec![FaceKind::Fluid; g.n_ufaces()];
+        let mut v_kind = vec![FaceKind::Fluid; g.n_vfaces()];
+        let mut w_kind = vec![FaceKind::Fluid; g.n_wfaces()];
+        for k in 0..nz {
+            for j in 0..ny {
+                for i in 1..nx {
+                    let f = g.uface(k, j, i);
+                    u_kind[f] = kind(cut.a_u[f]);
+                }
+            }
+            for j in 1..ny {
+                for i in 0..nx {
+                    let f = g.vface(k, j, i);
+                    v_kind[f] = kind(cut.a_v[f]);
+                }
+            }
+        }
+        let w_range = if periodic { 0..nz + 1 } else { 1..nz };
+        for k in w_range {
+            for j in 0..ny {
+                for i in 0..nx {
+                    let f = g.wface(k, j, i);
+                    w_kind[f] = kind(cut.a_w[f]);
+                }
+            }
+        }
+        Ok(Self {
+            grid: g,
+            periodic_z: periodic,
+            cell_fluid,
+            u_kind,
+            v_kind,
+            w_kind,
+            u_ghosts: Vec::new(),
+            v_ghosts: Vec::new(),
+            w_ghosts: Vec::new(),
+            anchor,
+            fluid_cells,
+            cut: Some(cut),
+        })
+    }
+
+    pub(super) fn lattice(&self) -> Lattice {
+        Lattice {
+            g: self.grid,
+            periodic_z: self.periodic_z,
+        }
+    }
+
+    /// Aperture of the face of component `c` at lattice `p` (1 without a
+    /// cut geometry), `None` outside the grid.
+    pub(super) fn aperture(&self, c: usize, p: [i64; 3]) -> Option<f64> {
+        let f = self.lattice().face(c, p)?;
+        Some(match c {
+            0 => self.a_u(f),
+            1 => self.a_v(f),
+            _ => self.a_w(f),
+        })
+    }
+
+    /// The momentum control volume of the unknown face of component `c` at
+    /// `p`: its face apertures are the averages of the two adjacent cells'
+    /// (the own-direction faces at the cell centres average the face's and
+    /// its own-direction neighbours' apertures), its wall closes them, its
+    /// wall distance is the face centre's signed distance moved to the
+    /// fluid part's centre, `φ + ½h(1 − α)`, floored.
+    pub(super) fn cv_geometry(&self, c: usize, p: [i64; 3]) -> CvGeometry {
+        let g = self.grid;
+        let h = [g.dx, g.dy, g.dz];
+        let area = [g.dy * g.dz, g.dx * g.dz, g.dx * g.dy];
+        let mut e = |d: usize| {
+            let mut v = [0i64; 3];
+            v[d] = 1;
+            v
+        };
+        let add =
+            |a: [i64; 3], b: [i64; 3], s: i64| [a[0] + s * b[0], a[1] + s * b[1], a[2] + s * b[2]];
+        let ec = e(c);
+        let cell_minus = add(p, ec, -1);
+        let cell_plus = p;
+        let alpha = self.aperture(c, p).unwrap_or(1.0);
+        let mut ap = [[1.0; 2]; 3];
+        for d in 0..3 {
+            let ed = e(d);
+            if d == c {
+                let am = self.aperture(c, add(p, ec, -1)).unwrap_or(alpha);
+                let apl = self.aperture(c, add(p, ec, 1)).unwrap_or(alpha);
+                ap[d] = [0.5 * (am + alpha), 0.5 * (alpha + apl)];
+            } else {
+                let minus = 0.5
+                    * (self.aperture(d, cell_minus).unwrap_or(1.0)
+                        + self.aperture(d, cell_plus).unwrap_or(1.0));
+                let plus = 0.5
+                    * (self.aperture(d, add(cell_minus, ed, 1)).unwrap_or(1.0)
+                        + self.aperture(d, add(cell_plus, ed, 1)).unwrap_or(1.0));
+                ap[d] = [minus, plus];
+            }
+        }
+        let mut wall = [0.0; 3];
+        for d in 0..3 {
+            wall[d] = -(ap[d][1] - ap[d][0]) * area[d];
+        }
+        let h_min = g.dx.min(g.dy).min(g.dz);
+        let phi_face = self.cut.as_ref().map_or(h_min, |cut| {
+            let f = self
+                .lattice()
+                .face(c, p)
+                .expect("unknown face inside the grid");
+            match c {
+                0 => cut.d_u[f],
+                1 => cut.d_v[f],
+                _ => cut.d_w[f],
+            }
+        });
+        let distance = (phi_face + 0.5 * h[c] * (1.0 - alpha)).max(DISTANCE_FLOOR * h_min);
+        CvGeometry {
+            alpha,
+            ap,
+            wall,
+            distance,
+        }
+    }
+
+    /// The surface velocity component `c` at the foot of the normal from
+    /// the face centre.
+    pub fn surface_velocity_at(&self, body: &Body, x: [f64; 3], c: usize, t: f64) -> f64 {
+        let g = self.grid;
+        let eps = 1e-6 * g.dx.min(g.dy).min(g.dz);
+        let s = body.phi(x[0], x[1], x[2], t);
+        let (n1, n2, n3) = body.normal(x[0], x[1], x[2], t, eps);
+        let v = body.surface_velocity(x[0] - s * n1, x[1] - s * n2, x[2] - s * n3, t);
+        [v.0, v.1, v.2][c]
+    }
+
+    /// The volume fluxes of the surface velocity through every cell's wall
+    /// into the body, `U_b·W_c` (zero for a body at rest; the porous
+    /// manufactured surface's flux otherwise), made compatible: the net
+    /// flux (the quadrature's defect on a closed surface — a rigid
+    /// translation's is zero by closure) is redistributed over the wall
+    /// cells by wall area, as the binary wall's ghost fluxes are. Returns
+    /// the table and the correction (flux per unit wall area).
+    pub fn wall_flux_table(&self, body: &Body, t: f64) -> (Vec<f64>, f64) {
+        let mut table = vec![0.0; self.grid.cells()];
+        let Some(cut) = self.cut.as_ref() else {
+            return (table, 0.0);
+        };
+        let (mut net, mut area) = (0.0, 0.0);
+        for idx in 0..table.len() {
+            if !self.cell_fluid[idx] {
+                continue;
+            }
+            let w = cut.wall[idx];
+            let a = (w[0] * w[0] + w[1] * w[1] + w[2] * w[2]).sqrt();
+            if a == 0.0 {
+                continue;
+            }
+            table[idx] = self.wall_flux(body, idx, t);
+            net += table[idx];
+            area += a;
+        }
+        let correction = if area > 0.0 { net / area } else { 0.0 };
+        if correction != 0.0 {
+            for (idx, w) in cut.wall.iter().enumerate() {
+                let a = (w[0] * w[0] + w[1] * w[1] + w[2] * w[2]).sqrt();
+                table[idx] -= correction * a;
+            }
+        }
+        (table, correction)
+    }
+
+    /// The volume flux of the surface velocity through a cell's wall into
+    /// the body, `U_b·W_c`, uncorrected. Zero without a cut geometry.
+    pub fn wall_flux(&self, body: &Body, idx: usize, t: f64) -> f64 {
+        let Some(cut) = self.cut.as_ref() else {
+            return 0.0;
+        };
+        let w = cut.wall[idx];
+        if w == [0.0; 3] {
+            return 0.0;
+        }
+        let g = self.grid;
+        let (k, j, i) = g.kji(idx);
+        let x = [
+            (i as f64 + 0.5) * g.dx,
+            (j as f64 + 0.5) * g.dy,
+            (k as f64 + 0.5) * g.dz,
+        ];
+        let mut flux = 0.0;
+        for (c, wc) in w.iter().enumerate() {
+            flux += self.surface_velocity_at(body, x, c, t) * wc;
+        }
+        flux
+    }
+
+    /// The cut-cell load route: the force on the body from the operators
+    /// themselves — `Σ_c p_c W_c` over the cells plus the implicit wall
+    /// shear `Σ_f μ A_w (u_f − U_b)/d_f` over the unknown faces. `None`
+    /// without a cut geometry.
+    pub fn cut_wall_force(&self, body: &Body, f: &Field, mu: f64, t: f64) -> Option<[f64; 3]> {
+        let cut = self.cut.as_ref()?;
+        let g = self.grid;
+        let (nx, ny, nz) = (g.nx, g.ny, g.nz);
+        let mut force = [0.0; 3];
+        for (idx, w) in cut.wall.iter().enumerate() {
+            if self.cell_fluid[idx] {
+                for c in 0..3 {
+                    force[c] += f.p[idx] * w[c];
+                }
+            }
+        }
+        let lat = self.lattice();
+        let values: [&[f64]; 3] = [&f.u, &f.v, &f.w];
+        let w_range = if self.periodic_z { 0..nz } else { 1..nz };
+        for c in 0..3 {
+            let (ir, jr, kr) = match c {
+                0 => (1..nx, 0..ny, 0..nz),
+                1 => (0..nx, 1..ny, 0..nz),
+                _ => (0..nx, 0..ny, w_range.clone()),
+            };
+            for k in kr {
+                for j in jr.clone() {
+                    for i in ir.clone() {
+                        let p = [i as i64, j as i64, k as i64];
+                        let idx = lat.face(c, p).expect("face");
+                        let kind = match c {
+                            0 => self.u_kind[idx],
+                            1 => self.v_kind[idx],
+                            _ => self.w_kind[idx],
+                        };
+                        if kind != FaceKind::Fluid {
+                            continue;
+                        }
+                        let cv = self.cv_geometry(c, p);
+                        let a_w = (cv.wall[0] * cv.wall[0]
+                            + cv.wall[1] * cv.wall[1]
+                            + cv.wall[2] * cv.wall[2])
+                            .sqrt();
+                        if a_w == 0.0 {
+                            continue;
+                        }
+                        let ub = self.surface_velocity_at(body, lat.face_position(c, p), c, t);
+                        force[c] += mu * a_w * (values[c][idx] - ub) / cv.distance;
+                    }
+                }
+            }
+        }
+        Some(force)
+    }
+}

crates/specialized/rtx-cfd/src/solvers/incompressible/embedded3/loads.rs

@@ -379,6 +379,7 @@ impl Mask {
                     if !self.is_fluid_cell(idx) {
                         continue;
                     }
+                    let dv = dv * self.vol(idx);
                     let (fu0, fu1) = (g.uface(k, j, i), g.uface(k, j, i + 1));
                     let (fv0, fv1) = (g.vface(k, j, i), g.vface(k, j + 1, i));
                     let (fw0, fw1) = (g.wface(k, j, i), g.wface(k + 1, j, i));

crates/specialized/rtx-cfd/src/solvers/incompressible/embedded3/mod.rs

@@ -8,6 +8,7 @@
 
 pub mod body;
 pub mod cut;
+pub mod cutwall;
 pub mod field;
 pub mod grid;
 pub mod loads;

crates/specialized/rtx-cfd/src/solvers/incompressible/embedded3/step/cut_predictor.rs

@@ -0,0 +1,215 @@
+//! The predictor on the apertured cut-cell wall (`cutwall.rs`): one
+//! routine for the three components, addressed on the face lattice. The
+//! momentum control volume of an unknown face is `V_u = α h A` closed by
+//! the wall; its faces carry the mass fluxes averaged from the two adjacent
+//! cells (the 2D face velocities when every aperture is 1), upwind plus the
+//! TVD correction as the 2D predictor, apertured diffusion, the pressure
+//! force `−(p₊ − p₋) α A` (the projection's gradient), the wall's momentum
+//! flux `m_w U_b` with `m_w = −Σ m_f` (so a uniform field stays uniform),
+//! and the implicit wall shear `μ A_w (u − U_b)/d_f`; the time derivative
+//! carries the inertia floor.
+
+use super::{Side, Solver};
+use crate::solvers::incompressible::embedded3::cutwall::INERTIA_FLOOR;
+use crate::solvers::incompressible::embedded3::field::Field;
+use crate::solvers::incompressible::simple::ConvectionScheme;
+
+impl Solver {
+    /// The three components' predictors on the unknown faces; the
+    /// prescribed faces keep their imposed values.
+    pub(super) fn cut_predictor(&self, field: &mut Field, dt: f64, t_old: f64) {
+        let mask = self.mask.as_ref().expect("cut mask");
+        let g = field.grid;
+        let (nx, ny, nz) = (g.nx, g.ny, g.nz);
+        let periodic = self.params.boundaries.periodic_z();
+        let lat = mask.lattice();
+        let w_range = if periodic { 0..nz } else { 1..nz };
+        for c in 0..3 {
+            let (ir, jr, kr) = match c {
+                0 => (1..nx, 0..ny, 0..nz),
+                1 => (0..nx, 1..ny, 0..nz),
+                _ => (0..nx, 0..ny, w_range.clone()),
+            };
+            let mut updates = Vec::new();
+            for k in kr {
+                for j in jr.clone() {
+                    for i in ir.clone() {
+                        let fluid = match c {
+                            0 => self.u_is_fluid(k, j, i),
+                            1 => self.v_is_fluid(k, j, i),
+                            _ => self.w_is_fluid(k, j, i),
+                        };
+                        if !fluid {
+                            continue;
+                        }
+                        let p = [i as i64, j as i64, k as i64];
+                        let idx = lat.face(c, p).expect("face");
+                        updates.push((idx, self.cut_face_update(field, c, p, dt, t_old)));
+                    }
+                }
+            }
+            let out: &mut Vec<f64> = match c {
+                0 => &mut field.u,
+                1 => &mut field.v,
+                _ => &mut field.w,
+            };
+            for (idx, val) in updates {
+                out[idx] = val;
+            }
+        }
+        if periodic {
+            for j in 0..ny {
+                for i in 0..nx {
+                    field.w[g.wface(nz, j, i)] = field.w[g.wface(0, j, i)];
+                }
+            }
+        }
+    }
+
+    /// The predicted value of the unknown face of component `c` at lattice
+    /// `p` from the old field.
+    #[allow(clippy::too_many_lines)]
+    fn cut_face_update(&self, field: &Field, c: usize, p: [i64; 3], dt: f64, t_old: f64) -> f64 {
+        let mask = self.mask.as_ref().expect("cut mask");
+        let body = self.body.as_ref().expect("body");
+        let g = field.grid;
+        let h = [g.dx, g.dy, g.dz];
+        let n = [g.nx as f64, g.ny as f64, g.nz as f64];
+        let area = [g.dy * g.dz, g.dx * g.dz, g.dx * g.dy];
+        let rho = self.fluid.density;
+        let mu = self.fluid.viscosity;
+        let b = self.params.boundaries;
+        let sides = [[b.x0, b.x1], [b.y0, b.y1], [b.z0, b.z1]];
+        let scheme = self.params.convection_scheme;
+        let lat = mask.lattice();
+        let old: [&[f64]; 3] = [&field.u_old, &field.v_old, &field.w_old];
+        let e = |d: usize| {
+            let mut v = [0i64; 3];
+            v[d] = 1;
+            v
+        };
+        let add =
+            |a: [i64; 3], b: [i64; 3], s: i64| [a[0] + s * b[0], a[1] + s * b[1], a[2] + s * b[2]];
+        // The old value of a face of component `cc` at `q`, `None` outside.
+        let val = |cc: usize, q: [i64; 3]| lat.face(cc, q).map(|f| old[cc][f]);
+        let ap = |cc: usize, q: [i64; 3]| mask.aperture(cc, q);
+        let cv = mask.cv_geometry(c, p);
+        let x = lat.face_position(c, p);
+        let u0 = val(c, p).expect("the face");
+        let ec = e(c);
+        let cell_minus = add(p, ec, -1);
+        let cell_plus = p;
+        let ub = mask.surface_velocity_at(body, x, c, t_old);
+
+        let mut mass_out = 0.0;
+        let mut conv = 0.0;
+        let mut diff = 0.0;
+        for d in 0..3 {
+            let ed = e(d);
+            let a_d = area[d];
+            // Neighbouring faces of this component along d (None beyond a wall).
+            let up1 = val(c, add(p, ed, 1));
+            let up2 = val(c, add(p, ed, 2));
+            let dn1 = val(c, add(p, ed, -1));
+            let dn2 = val(c, add(p, ed, -2));
+            // The control volume's mass fluxes through its plus / minus faces
+            // along d: the averages of the two adjacent cells' face fluxes
+            // (own direction: the face's and its neighbours' fluxes).
+            let (m_plus, m_minus) = if d == c {
+                let f_up = ap(c, add(p, ec, 1)).unwrap_or(cv.alpha) * up1.unwrap_or(u0);
+                let f_dn = ap(c, add(p, ec, -1)).unwrap_or(cv.alpha) * dn1.unwrap_or(u0);
+                let f0 = cv.alpha * u0;
+                (0.5 * (f0 + f_up) * a_d, 0.5 * (f_dn + f0) * a_d)
+            } else {
+                let flux = |q: [i64; 3]| ap(d, q).unwrap_or(1.0) * val(d, q).unwrap_or(0.0);
+                (
+                    0.5 * (flux(add(cell_minus, ed, 1)) + flux(add(cell_plus, ed, 1))) * a_d,
+                    0.5 * (flux(cell_minus) + flux(cell_plus)) * a_d,
+                )
+            };
+            mass_out += m_plus - m_minus;
+            // Beyond a domain side along d: the boundary value on a Velocity
+            // side, the face's own value (mirror) otherwise.
+            let beyond = |plus: bool| {
+                let side = sides[d][usize::from(plus)];
+                if side == Side::Velocity {
+                    let mut xb = x;
+                    xb[d] = if plus { n[d] * h[d] } else { 0.0 };
+                    [
+                        self.boundary(xb[0], xb[1], xb[2], t_old).0,
+                        self.boundary(xb[0], xb[1], xb[2], t_old).1,
+                        self.boundary(xb[0], xb[1], xb[2], t_old).2,
+                    ][c]
+                } else {
+                    u0
+                }
+            };
+            // Convection through the plus face.
+            let (u_plus, delta_plus) = match up1 {
+                Some(un) => {
+                    let delta = if scheme == ConvectionScheme::Upwind {
+                        0.0
+                    } else if m_plus >= 0.0 {
+                        scheme.face_correction(dn1, u0, un)
+                    } else {
+                        scheme.face_correction(up2, un, u0)
+                    };
+                    (Self::upwind(m_plus, u0, un), delta)
+                }
+                None => (Self::upwind(m_plus, u0, beyond(true)), 0.0),
+            };
+            let (u_minus, delta_minus) = match dn1 {
+                Some(ud) => {
+                    let delta = if scheme == ConvectionScheme::Upwind {
+                        0.0
+                    } else if m_minus >= 0.0 {
+                        scheme.face_correction(dn2, ud, u0)
+                    } else {
+                        scheme.face_correction(up1, u0, ud)
+                    };
+                    (Self::upwind(m_minus, ud, u0), delta)
+                }
+                None => (Self::upwind(m_minus, beyond(false), u0), 0.0),
+            };
+            conv += m_plus * (u_plus + delta_plus) - m_minus * (u_minus + delta_minus);
+            // Diffusion through the plus / minus faces.
+            let (g_minus, g_plus) = (cv.ap[d][0], cv.ap[d][1]);
+            diff += match up1 {
+                Some(un) => mu * g_plus * a_d * (un - u0) / h[d],
+                None => {
+                    if sides[d][1] == Side::Velocity {
+                        mu * g_plus * a_d * (beyond(true) - u0) / (0.5 * h[d])
+                    } else {
+                        0.0
+                    }
+                }
+            };
+            diff -= match dn1 {
+                Some(ud) => mu * g_minus * a_d * (u0 - ud) / h[d],
+                None => {
+                    if sides[d][0] == Side::Velocity {
+                        mu * g_minus * a_d * (u0 - beyond(false)) / (0.5 * h[d])
+                    } else {
+                        0.0
+                    }
+                }
+            };
+        }
+        // The wall's momentum flux closes the mass balance exactly.
+        conv -= mass_out * ub;
+        let p_plus = lat.cell(cell_plus).map_or(0.0, |ci| field.p[ci]);
+        let p_minus = lat.cell(cell_minus).map_or(0.0, |ci| field.p[ci]);
+        let pressure = -(p_plus - p_minus) * cv.alpha * area[c];
+        let v_u = cv.alpha * h[c] * area[c];
+        let source = self.momentum_source.as_ref().map_or(0.0, |f| {
+            let s = f(x[0], x[1], x[2], t_old);
+            [s.0, s.1, s.2][c] * v_u
+        });
+        let a_w =
+            (cv.wall[0] * cv.wall[0] + cv.wall[1] * cv.wall[1] + cv.wall[2] * cv.wall[2]).sqrt();
+        let shear = mu * a_w / cv.distance;
+        let v_eff = cv.alpha.max(INERTIA_FLOOR) * h[c] * area[c];
+        let inertia = rho * v_eff / dt;
+        (inertia * u0 - conv + diff + pressure + source + shear * ub) / (inertia + shear)
+    }
+}

crates/specialized/rtx-cfd/src/solvers/incompressible/embedded3/step/mod.rs

@@ -4,6 +4,7 @@
 //! `dz = 1`) every number is the 2D solver's. The fluid predicates are the
 //! wall's hooks (item 9).
 
+mod cut_predictor;
 #[cfg(feature = "cuda")]
 pub mod device;
 mod predictor;
@@ -110,6 +111,7 @@ pub struct Solver {
     body: Option<Body>,
     mask: Option<Mask>,
     last_ghost_correction: f64,
+    wall_fluxes: Vec<f64>,
     pcg_cache: PcgCache,
     time: f64,
     initialized: bool,
@@ -137,6 +139,7 @@ impl Solver {
             body: None,
             mask: None,
             last_ghost_correction: 0.0,
+            wall_fluxes: Vec::new(),
             pcg_cache: PcgCache::default(),
             time: 0.0,
             initialized: false,
@@ -174,7 +177,8 @@ impl Solver {
         self.mask.as_ref()
     }
 
-    /// The last step's ghost compatibility correction.
+    /// The last step's compatibility correction: the binary wall's shared
+    /// ghost flux correction, or the cut wall's wall-flux correction.
     #[must_use]
     pub fn ghost_correction(&self) -> f64 {
         self.last_ghost_correction
@@ -226,6 +230,36 @@ impl Solver {
             .is_none_or(|m| m.is_fluid_cell(m.grid().cell(k, j, i)))
     }
 
+    // The apertures (1 without a cut geometry).
+    #[inline]
+    pub(super) fn au(&self, k: usize, j: usize, i: usize) -> f64 {
+        self.mask
+            .as_ref()
+            .map_or(1.0, |m| m.a_u(m.grid().uface(k, j, i)))
+    }
+    #[inline]
+    pub(super) fn av(&self, k: usize, j: usize, i: usize) -> f64 {
+        self.mask
+            .as_ref()
+            .map_or(1.0, |m| m.a_v(m.grid().vface(k, j, i)))
+    }
+    #[inline]
+    pub(super) fn aw(&self, k: usize, j: usize, i: usize) -> f64 {
+        self.mask
+            .as_ref()
+            .map_or(1.0, |m| m.a_w(m.grid().wface(k, j, i)))
+    }
+    /// The surface velocity's compatible flux through the cell's wall at
+    /// the step's new time (cut wall only; the table is rebuilt per step).
+    #[inline]
+    pub(super) fn wall_flux(&self, idx: usize) -> f64 {
+        self.wall_fluxes[idx]
+    }
+    #[inline]
+    pub(super) fn has_cut(&self) -> bool {
+        self.mask.as_ref().is_some_and(|m| m.cut().is_some())
+    }
+
     pub(super) fn upwind(face_velocity: f64, upstream: f64, downstream: f64) -> f64 {
         if face_velocity >= 0.0 {
             upstream
@@ -283,7 +317,10 @@ impl Solver {
         let (nx, ny, nz) = (g.nx, g.ny, g.nz);
         let b = self.params.boundaries;
         let periodic = b.periodic_z();
-        for k in 0..nz {
+        if self.has_cut() {
+            self.cut_predictor(field, dt, t_old);
+        }
+        for k in (0..nz).filter(|_| !self.has_cut()) {
             for j in 0..ny {
                 for i in 1..nx {
                     if !self.u_is_fluid(k, j, i) {
@@ -295,7 +332,7 @@ impl Solver {
                 }
             }
         }
-        for k in 0..nz {
+        for k in (0..nz).filter(|_| !self.has_cut()) {
             for j in 1..ny {
                 for i in 0..nx {
                     if !self.v_is_fluid(k, j, i) {
@@ -308,7 +345,7 @@ impl Solver {
             }
         }
         let k_range = if periodic { 0..nz } else { 1..nz };
-        for k in k_range {
+        for k in k_range.filter(|_| !self.has_cut()) {
             for j in 0..ny {
                 for i in 0..nx {
                     if !self.w_is_fluid(k, j, i) {
@@ -365,15 +402,15 @@ impl Solver {
         let t = self.time;
         if let Some(body) = &self.body {
             if self.mask.is_none() {
-                assert_eq!(
+                let mask = match self.params.wall_scheme {
-                    self.params.wall_scheme,
+                    WallScheme::GhostBinary => {
-                    WallScheme::GhostBinary,
-                    "item 10 brings CutCell"
-                );
-                self.mask = Some(
                         Mask::build(body, field.grid, t, self.params.boundaries)
-                        .expect("embedded mask"),
+                    }
-                );
+                    WallScheme::CutCell => {
+                        Mask::build_cut(body, field.grid, t, self.params.boundaries)
+                    }
+                };
+                self.mask = Some(mask.expect("embedded mask"));
             }
         }
         self.apply_boundary_normals(field, t);
@@ -398,6 +435,14 @@ impl Solver {
         self.momentum_predictor(field, dt, t_old);
         self.apply_boundary_normals(field, t_new);
         field.copy_to_starred();
+        let mut cut_correction = None;
+        if let (Some(body), Some(mask)) = (&self.body, &self.mask) {
+            if mask.cut().is_some() {
+                let (table, correction) = mask.wall_flux_table(body, t_new);
+                self.wall_fluxes = table;
+                cut_correction = Some(correction);
+            }
+        }
         let mut total = 0;
         let mut final_residual = f64::INFINITY;
         let mut poisson_iterations = 0;
@@ -414,8 +459,8 @@ impl Solver {
         }
         // Ghost faces follow the corrected field (the next step's stencil data).
         if let (Some(body), Some(mask)) = (&self.body, &self.mask) {
-            self.last_ghost_correction =
+            let imposed = mask.impose(body, &mut field.u, &mut field.v, &mut field.w, t_new);
-                mask.impose(body, &mut field.u, &mut field.v, &mut field.w, t_new);
+            self.last_ghost_correction = cut_correction.unwrap_or(imposed);
         }
         self.time = t_new;
         StepResult {

crates/specialized/rtx-cfd/src/solvers/incompressible/embedded3/step/projection.rs

@@ -39,42 +39,42 @@ impl Solver {
                             extra += ae_outlet;
                         }
                     } else if self.u_is_fluid(k, j, i + 1) {
-                        problem.ae[idx] = ae_interior;
+                        problem.ae[idx] = ae_interior * self.au(k, j, i + 1);
                     }
                     if i == 0 {
                         if b.x0 == outlet {
                             extra += ae_outlet;
                         }
                     } else if self.u_is_fluid(k, j, i) {
-                        problem.aw[idx] = ae_interior;
+                        problem.aw[idx] = ae_interior * self.au(k, j, i);
                     }
                     if j + 1 == ny {
                         if b.y1 == outlet {
                             extra += an_outlet;
                         }
                     } else if self.v_is_fluid(k, j + 1, i) {
-                        problem.an[idx] = an_interior;
+                        problem.an[idx] = an_interior * self.av(k, j + 1, i);
                     }
                     if j == 0 {
                         if b.y0 == outlet {
                             extra += an_outlet;
                         }
                     } else if self.v_is_fluid(k, j, i) {
-                        problem.as_[idx] = an_interior;
+                        problem.as_[idx] = an_interior * self.av(k, j, i);
                     }
                     if k + 1 == nz && !periodic {
                         if b.z1 == outlet {
                             extra += at_outlet;
                         }
                     } else if self.w_is_fluid((k + 1) % nz, j, i) {
-                        problem.at[idx] = at_interior;
+                        problem.at[idx] = at_interior * self.aw((k + 1) % nz, j, i);
                     }
                     if k == 0 && !periodic {
                         if b.z0 == outlet {
                             extra += at_outlet;
                         }
                     } else if self.w_is_fluid(k, j, i) {
-                        problem.ab[idx] = at_interior;
+                        problem.ab[idx] = at_interior * self.aw(k, j, i);
                     }
                     problem.extra_diag[idx] = extra;
                 }
@@ -246,6 +246,7 @@ impl Solver {
         let g = field.grid;
         let (nx, ny, nz, dx, dy, dz) = (g.nx, g.ny, g.nz, g.dx, g.dy, g.dz);
         let rho = self.fluid.density;
+        let cut = self.has_cut();
         let mut source_scale = 0.0;
         for k in 0..nz {
             for j in 0..ny {
@@ -255,15 +256,19 @@ impl Solver {
                         field.sp[idx] = 0.0;
                         continue;
                     }
-                    let divergence_flux = rho
+                    let mut divergence_flux = rho
-                        * ((field.u_star[g.uface(k, j, i + 1)] - field.u_star[g.uface(k, j, i)])
+                        * ((self.au(k, j, i + 1) * field.u_star[g.uface(k, j, i + 1)]
+                            - self.au(k, j, i) * field.u_star[g.uface(k, j, i)])
                             * (dy * dz)
-                            + (field.v_star[g.vface(k, j + 1, i)]
+                            + (self.av(k, j + 1, i) * field.v_star[g.vface(k, j + 1, i)]
-                                - field.v_star[g.vface(k, j, i)])
+                                - self.av(k, j, i) * field.v_star[g.vface(k, j, i)])
                                 * (dx * dz)
-                            + (field.w_star[g.wface(k + 1, j, i)]
+                            + (self.aw(k + 1, j, i) * field.w_star[g.wface(k + 1, j, i)]
-                                - field.w_star[g.wface(k, j, i)])
+                                - self.aw(k, j, i) * field.w_star[g.wface(k, j, i)])
                                 * (dx * dy));
+                    if cut {
+                        divergence_flux += rho * self.wall_flux(idx);
+                    }
                     field.sp[idx] = -divergence_flux;
                     source_scale += divergence_flux.abs();
                 }
@@ -315,6 +320,7 @@ impl Solver {
         let g = field.grid;
         let (nx, ny, nz, dx, dy, dz) = (g.nx, g.ny, g.nz, g.dx, g.dy, g.dz);
         let rho = self.fluid.density;
+        let cut = self.has_cut();
         let b = self.params.boundaries;
         let outlet = Side::PressureOutlet;
         let periodic = b.periodic_z();
@@ -402,12 +408,20 @@ impl Solver {
                     if !self.cell_is_fluid(k, j, i) {
                         continue;
                     }
-                    let divergence_flux = rho
+                    let idx = g.cell(k, j, i);
-                        * ((field.u[g.uface(k, j, i + 1)] - field.u[g.uface(k, j, i)]) * (dy * dz)
+                    let mut divergence_flux = rho
-                            + (field.v[g.vface(k, j + 1, i)] - field.v[g.vface(k, j, i)])
+                        * ((self.au(k, j, i + 1) * field.u[g.uface(k, j, i + 1)]
+                            - self.au(k, j, i) * field.u[g.uface(k, j, i)])
+                            * (dy * dz)
+                            + (self.av(k, j + 1, i) * field.v[g.vface(k, j + 1, i)]
+                                - self.av(k, j, i) * field.v[g.vface(k, j, i)])
                                 * (dx * dz)
-                            + (field.w[g.wface(k + 1, j, i)] - field.w[g.wface(k, j, i)])
+                            + (self.aw(k + 1, j, i) * field.w[g.wface(k + 1, j, i)]
+                                - self.aw(k, j, i) * field.w[g.wface(k, j, i)])
                                 * (dx * dy));
+                    if cut {
+                        divergence_flux += rho * self.wall_flux(idx);
+                    }
                     mass_imbalance += divergence_flux.abs();
                 }
             }

crates/specialized/rtx-cfd/src/solvers/incompressible/embedded3/wall.rs

@@ -10,6 +10,7 @@
 
 use super::Grid;
 use super::body::Body;
+use super::cut::CutGeometry;
 use super::step::{Boundaries, Side};
 
 #[derive(Debug, Clone, Copy, PartialEq, Eq)]
@@ -45,7 +46,7 @@ pub(crate) struct StencilNode {
 }
 
 #[derive(Debug, Clone)]
-struct Ghost {
+pub(super) struct Ghost {
     idx: usize,
     x: f64,
     y: f64,
@@ -62,17 +63,20 @@ struct Ghost {
 
 #[derive(Clone)]
 pub struct Mask {
-    grid: Grid,
+    pub(super) grid: Grid,
-    periodic_z: bool,
+    pub(super) periodic_z: bool,
-    cell_fluid: Vec<bool>,
+    pub(super) cell_fluid: Vec<bool>,
-    u_kind: Vec<FaceKind>,
+    pub(super) u_kind: Vec<FaceKind>,
-    v_kind: Vec<FaceKind>,
+    pub(super) v_kind: Vec<FaceKind>,
-    w_kind: Vec<FaceKind>,
+    pub(super) w_kind: Vec<FaceKind>,
-    u_ghosts: Vec<Ghost>,
+    pub(super) u_ghosts: Vec<Ghost>,
-    v_ghosts: Vec<Ghost>,
+    pub(super) v_ghosts: Vec<Ghost>,
-    w_ghosts: Vec<Ghost>,
+    pub(super) w_ghosts: Vec<Ghost>,
-    anchor: usize,
+    pub(super) anchor: usize,
-    fluid_cells: usize,
+    pub(super) fluid_cells: usize,
+    /// The cut geometry of the apertured wall (`WallScheme::CutCell`,
+    /// `cutwall.rs`); `None` on the binary ghost wall.
+    pub(super) cut: Option<CutGeometry>,
 }
 
 /// The z lattice position of a query: the lower plane index, the upper
@@ -479,9 +483,38 @@ impl Mask {
             w_ghosts,
             anchor,
             fluid_cells,
+            cut: None,
         })
     }
 
+    /// The cut geometry (apertured wall only).
+    #[must_use]
+    pub fn cut(&self) -> Option<&CutGeometry> {
+        self.cut.as_ref()
+    }
+    /// Fluid area fraction of a u / v / w face (1 on the binary wall).
+    #[inline]
+    #[must_use]
+    pub fn a_u(&self, idx: usize) -> f64 {
+        self.cut.as_ref().map_or(1.0, |c| c.a_u[idx])
+    }
+    #[inline]
+    #[must_use]
+    pub fn a_v(&self, idx: usize) -> f64 {
+        self.cut.as_ref().map_or(1.0, |c| c.a_v[idx])
+    }
+    #[inline]
+    #[must_use]
+    pub fn a_w(&self, idx: usize) -> f64 {
+        self.cut.as_ref().map_or(1.0, |c| c.a_w[idx])
+    }
+    /// Fluid volume fraction of a cell (1 on the binary wall).
+    #[inline]
+    #[must_use]
+    pub fn vol(&self, idx: usize) -> f64 {
+        self.cut.as_ref().map_or(1.0, |c| c.vol[idx])
+    }
+
     #[inline]
     #[must_use]
     pub fn is_fluid_cell(&self, idx: usize) -> bool {

crates/specialized/rtx-cfd/tests/embedded3_embedded_mms.rs

@@ -1,14 +1,20 @@
-//! embedded3 gate 9a: the manufactured solution with an embedded sphere
+//! embedded3 gates 9a and 10: the manufactured solution with an embedded
-//! (centre (0.6, 0.45, 0.5), r 0.2, off-centre so the exact force is not
+//! sphere (centre (0.6, 0.45, 0.5), r 0.2, off-centre so the exact force is
-//! zero by symmetry) carrying the exact field as its surface velocity, on
+//! not zero by symmetry) carrying the exact field as its surface velocity,
-//! the binary ghost wall. The velocity error falls at the scheme's order,
+//! on the binary ghost wall (item 9) and the apertured cut-cell wall (item
-//! every fluid cell is divergence-free, the compatibility correction
+//! 10). The velocity error falls at the scheme's order, every fluid cell
-//! shrinks, and both load routes converge to the exact surface integral of
+//! is divergence-free (apertured, with the porous surface's flux, on the
-//! the manufactured stress (the control-volume route measures F − M with M
+//! cut wall), the compatibility correction shrinks, and both load routes
-//! the momentum flux through the porous manufactured surface).
+//! converge to the exact surface integral of the manufactured stress (the
+//! control-volume route measures F − M with M the momentum flux through
+//! the porous manufactured surface). Item 10's gate: the cut wall's errors
+//! are at most the binary wall's at every n, its loads within 10 % at the
+//! finest rung.
 
 use rtx_cfd::solvers::incompressible::ConvectionScheme;
-use rtx_cfd::solvers::incompressible::embedded3::{Body, Field, Fluid, Grid, Parameters, Solver};
+use rtx_cfd::solvers::incompressible::embedded3::{
+    Body, FaceKind, Field, Fluid, Grid, Parameters, Solver, WallScheme,
+};
 use std::f64::consts::PI;
 
 const RHO: f64 = 1.0;
@@ -114,7 +120,7 @@ struct Measurement {
     force_cv: [f64; 3],
 }
 
-fn measure(n: usize) -> Measurement {
+fn measure(n: usize, scheme: WallScheme) -> Measurement {
     let h = 1.0 / n as f64;
     let dt = 0.4 * (h * h / (4.0 * MU / RHO)).min(h);
     let mut solver = Solver::new(
@@ -128,6 +134,7 @@ fn measure(n: usize) -> Measurement {
             corrector_steps: 2,
             tolerance: 1e-8,
             convection_scheme: ConvectionScheme::Upwind,
+            wall_scheme: scheme,
             ..Parameters::default()
         },
     );
@@ -140,9 +147,10 @@ fn measure(n: usize) -> Measurement {
     let g = Grid::cubic(n, n, n, h);
     let mut f = Field::new(g);
     solver.initialize(&mut f);
+    let mut last = solver.advance(&mut f, dt);
     for _ in 0..200_000 {
         let (bu, bv, bw) = (f.u.clone(), f.v.clone(), f.w.clone());
-        solver.advance(&mut f, dt);
+        last = solver.advance(&mut f, dt);
         let mut change = 0.0_f64;
         for (a, b) in
             f.u.iter()
@@ -157,7 +165,6 @@ fn measure(n: usize) -> Measurement {
         }
     }
     let mask = solver.mask().expect("mask");
-    use rtx_cfd::solvers::incompressible::embedded3::FaceKind;
     let (mut sq, mut vol) = (0.0, 0.0);
     let dv = h * h * h;
     for k in 0..n {
@@ -194,21 +201,53 @@ fn measure(n: usize) -> Measurement {
             }
         }
     }
+    let body = solver.body().expect("body");
+    let t = solver.time();
+    // The apertured divergence per unit volume, the porous surface's flux
+    // through the wall included (the plain divergence on the binary wall).
     let mut max_div = 0.0_f64;
+    let mut at_vol = 1.0;
+    let mut sum_flux = 0.0;
+    let (wall_fluxes, _) = mask.wall_flux_table(body, t);
     for k in 0..n {
         for j in 0..n {
             for i in 0..n {
-                if mask.is_fluid_cell(g.cell(k, j, i)) {
+                let idx = g.cell(k, j, i);
-                    let div = (f.u[g.uface(k, j, i + 1)] - f.u[g.uface(k, j, i)]) / h
+                if mask.is_fluid_cell(idx) {
-                        + (f.v[g.vface(k, j + 1, i)] - f.v[g.vface(k, j, i)]) / h
+                    let flux = (mask.a_u(g.uface(k, j, i + 1)) * f.u[g.uface(k, j, i + 1)]
-                        + (f.w[g.wface(k + 1, j, i)] - f.w[g.wface(k, j, i)]) / h;
+                        - mask.a_u(g.uface(k, j, i)) * f.u[g.uface(k, j, i)])
-                    max_div = max_div.max(div.abs());
+                        * h
+                        * h
+                        + (mask.a_v(g.vface(k, j + 1, i)) * f.v[g.vface(k, j + 1, i)]
+                            - mask.a_v(g.vface(k, j, i)) * f.v[g.vface(k, j, i)])
+                            * h
+                            * h
+                        + (mask.a_w(g.wface(k + 1, j, i)) * f.w[g.wface(k + 1, j, i)]
+                            - mask.a_w(g.wface(k, j, i)) * f.w[g.wface(k, j, i)])
+                            * h
+                            * h
+                        + wall_fluxes[idx];
+                    sum_flux += flux.abs();
+                    if (flux / (h * h * h)).abs() > max_div {
+                        max_div = (flux / (h * h * h)).abs();
+                        at_vol = mask.vol(idx);
                     }
                 }
             }
         }
-    let body = solver.body().expect("body");
+    }
-    let surface = mask.surface_force(body, &f, MU, solver.time(), 0.5 * h);
+    println!(
+        "    [{scheme:?} n {n}] max div {max_div:.2e} in a cell of fluid fraction {at_vol:.3e}; Σ|flux| {sum_flux:.2e}; last step residual {:.2e}",
+        last.final_residual
+    );
+    let surface = match scheme {
+        WallScheme::GhostBinary => mask.surface_force(body, &f, MU, t, 0.5 * h),
+        WallScheme::CutCell => rtx_cfd::solvers::incompressible::embedded3::SurfaceForce {
+            f: mask.cut_wall_force(body, &f, MU, t).expect("cut wall"),
+            samples: 0,
+            skipped: 0,
+        },
+    };
     let (i0, i1) = (n / 8, n - n / 8);
     let src = |x: f64, y: f64, z: f64| source3(x, y, z);
     let force_cv = mask.control_volume_force(&f, dt, RHO, MU, Some(&src), (i0, i1, i0, i1, i0, i1));
@@ -226,14 +265,21 @@ fn norm(a: [f64; 3]) -> f64 {
     (a[0] * a[0] + a[1] * a[1] + a[2] * a[2]).sqrt()
 }
 
-fn ladder(resolutions: &[usize]) {
+/// The velocity errors and the two routes' relative force errors per rung.
+struct Ladder {
+    errors: Vec<f64>,
+    surface: Vec<f64>,
+    cv: Vec<f64>,
+}
+
+fn ladder(resolutions: &[usize], scheme: WallScheme) -> Ladder {
     let (fe, m) = exact_force_and_flux();
     let f_scale = norm(fe);
     let fcv = [fe[0] - m[0], fe[1] - m[1], fe[2] - m[2]];
     println!(
-        "  exact force {fe:.5?}; momentum flux {m:.5?}; the control-volume route measures {fcv:.5?}"
+        "  {scheme:?}: exact force {fe:.5?}; momentum flux {m:.5?}; the control-volume route measures {fcv:.5?}"
     );
-    let ms: Vec<Measurement> = resolutions.iter().map(|&n| measure(n)).collect();
+    let ms: Vec<Measurement> = resolutions.iter().map(|&n| measure(n, scheme)).collect();
     let errors: Vec<f64> = ms.iter().map(|x| x.l2_velocity).collect();
     let mut se = Vec::new();
     let mut ce = Vec::new();
@@ -287,15 +333,57 @@ fn ladder(resolutions: &[usize]) {
         ce.windows(2).all(|w| w[1] < w[0]),
         "control-volume-route error not falling {ce:?}"
     );
+    Ladder {
+        errors,
+        surface: se,
+        cv: ce,
+    }
+}
+
+/// Item 10's comparison: the cut wall's velocity error at most the binary
+/// wall's at every rung; both routes within `load_bound` at the finest.
+fn compare(resolutions: &[usize], load_bound: f64) {
+    let ghost = ladder(resolutions, WallScheme::GhostBinary);
+    let cut = ladder(resolutions, WallScheme::CutCell);
+    for (k, &n) in resolutions.iter().enumerate() {
+        println!(
+            "  n = {n:3}  L2 u ghost {:.4e} cut {:.4e}  (ratio {:.3})",
+            ghost.errors[k],
+            cut.errors[k],
+            cut.errors[k] / ghost.errors[k]
+        );
+        assert!(
+            cut.errors[k] <= ghost.errors[k],
+            "cut-cell error above the binary wall's at n = {n}"
+        );
+    }
+    let last = resolutions.len() - 1;
+    assert!(
+        cut.surface[last] < load_bound && cut.cv[last] < load_bound,
+        "cut-cell loads at the finest rung: surface {:.3e}, control volume {:.3e} (bound {load_bound})",
+        cut.surface[last],
+        cut.cv[last]
+    );
 }
 
 #[test]
 fn embedded_sphere_recovers_the_manufactured_solution() {
-    ladder(&[12, 24]);
+    ladder(&[12, 24], WallScheme::GhostBinary);
+}
+
+#[test]
+fn cut_cell_wall_recovers_the_manufactured_solution() {
+    compare(&[12, 24], 0.2);
 }
 
 #[test]
 #[ignore = "the three-rung ladder to n = 48 (minutes on the host)"]
 fn embedded_sphere_three_rungs() {
-    ladder(&[12, 24, 48]);
+    ladder(&[12, 24, 48], WallScheme::GhostBinary);
+}
+
+#[test]
+#[ignore = "item 10's finest rung: the cut wall's loads within 10 % at n = 48"]
+fn cut_cell_three_rungs() {
+    compare(&[12, 24, 48], 0.1);
 }