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# Advection-Diffusion Model

PenguinTransportDiffusion.jl provides fixed and moving advection-diffusion models:

  • AdvDiffModelMono, AdvDiffModelDiph
  • MovingAdvDiffModelMono, MovingAdvDiffModelDiph

Continuous PDE

\[\partial_t T + \nabla \cdot (u T) = \nabla \cdot (D \nabla T) + f.\]

Discretization split

  • Diffusion + mass terms are delegated to PenguinDiffusion.
  • Advection operators and advection-side box BC injection are delegated to PenguinTransport.
  • This package composes both into one linear system over (\omega, \gamma) (mono) or (\omega_1,\gamma_1,\omega_2,\gamma_2) (diph).

Boundary handling

A single BorderConditions input is accepted by all models, then internally split:

  • Dirichlet / Neumann / Periodic are applied on the diffusion side.
  • Inflow / Outflow / Periodic are applied on the advection side.
  • Mixed unsupported boundary kinds raise an ArgumentError.

Interface condition

  • Mono diffusion interface coupling is optional via bc_interface_diff::Union{Nothing, PenguinBCs.Robin}.
  • Diph interface coupling is passed through ic::Union{Nothing,InterfaceConditions}.
  • A pure Neumann interface can introduce a \gamma-gauge nullspace; one \gamma row may need pinning in custom solves.

Time integration

  • Fixed geometry: assemble_unsteady_* / solve_unsteady! support :BE, :CN, numeric \theta.
  • Moving geometry: assemble_unsteady_*_moving! / solve_unsteady_moving! support the same values.
  • For \theta \neq 1, the explicit correction (1-\theta) A u^n is moved to the RHS.
  • Fixed mass contribution uses cell volumes V / \Delta t on \omega.
  • Moving assembly uses slab-reduced capacities from PenguinDiffusion and advection with relative interface speed (u_\gamma - w_\gamma) \cdot n_\gamma.