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Algorithms

1. Discrete Unknown Layouts

Mono:

x = [ φω ; φγ ]

Two-phase:

x = [ φω1 ; φγ1 ; φω2 ; φγ2 ]

The two-phase ordering is fixed in all public APIs.

2. Core Advection Blocks

CartesianOperators.jl provides cut-cell advection operators.

  • ω rows receive bulk transport + source + (unsteady) time terms
  • γ rows enforce local embedded-interface closure

For moving models, interface transport/closure uses the discrete relative interface coefficient assembled with (uγ-wγ).

3. Steady Mono Assembly

Block form:

\[\begin{bmatrix} A_{11} & A_{12}\\ A_{21} & A_{22} \end{bmatrix} \begin{bmatrix} \phi_\omega\\ \phi_\gamma \end{bmatrix} = \begin{bmatrix} b_\omega\\ b_\gamma \end{bmatrix}.\]

  • A11, A12: advection blocks
  • A21, A22: interface closure rows (Tγ=g inflow or Tγ=Tω continuity)
  • outer box BCs are then applied on ω rows

4. Unsteady Mono θ Assembly (Fixed)

Fixed geometry uses the usual θ method on ω rows:

\[\left(\frac{V}{\Delta t} + \theta A\right)\phi^{n+1} = \frac{V}{\Delta t}\phi^n - (1-\theta)A\phi^n + b.\]

solve_unsteady! reuses constant operators when matrix/RHS are time-invariant.

5. Unsteady Mono Moving-Slab Assembly

Moving geometry uses reduced slab volumes Vn, Vn1:

\[A_{11} = M_1 + A^{adv}_{11}\Psi_+, \qquad A_{12} = -(M_1-M_0) + A^{adv}_{12}\Psi_+,\]

\[b_\omega = (M_0 - A^{adv}_{11}\Psi_-)\phi_\omega^n - (A^{adv}_{12}\Psi_-)\phi_\gamma^n + b_{src}.\]

with M0=diag(Vn), M1=diag(Vn1).

The -(M1-M0) term carries the pure geometry sweep effect, so no extra standalone geometric flux term is added.

6. Steady/Unsteady Two-Phase (Fixed)

Per-phase ω blocks are assembled like mono.

γ1/γ2 rows are selected locally from discrete sign logic (κ1, κ2):

  • one inflow / one outflow: coupling row + outflow continuity row
  • both outflow: continuity on both
  • both inflow: rejected (ill-posed local closure)

7. Unsteady Two-Phase Moving-Slab

Same block philosophy as fixed two-phase, but with:

  • per-phase moving time terms from V1n/V1n1, V2n/V2n1
  • interface sign logic based on discrete relative coefficients κ1rel, κ2rel
  • both-inflow rejection based on those discrete signs

Outer box BC logic remains driven by bulk velocities u1ω, u2ω (not wγ).

8. Regularization / Inactive Rows

Inactive and halo rows are forced to identity equations.

This keeps matrix layouts stable and avoids singular systems as active topology changes.

9. Practical Notes

  • Centered() is less diffusive but can oscillate near sharp fronts.
  • Upwind1() is more robust/monotone but diffusive.
  • For moving models, expect first-order-in-time behavior with :BE; :CN gives less temporal damping but follows the same validated θ contract.