Diffusion Models and PDEs

Monophasic Model

In a domain Ω(t) with outer boundary ∂Ω, solve

\[\partial_t u = \nabla \cdot (D(x,t)\nabla u) + s(x,t)\]

with outer boundary conditions (Dirichlet/Neumann/Robin/Periodic depending on side).

For fixed geometry (DiffusionModelMono), Ω is time-independent. For moving geometry (MovingDiffusionModelMono), assembly uses a space-time slab.

Diphasic Model

For two phases separated by an embedded interface Γ:

\[\partial_t u_k = \nabla \cdot (D_k(x,t)\nabla u_k) + s_k(x,t), \quad k=1,2.\]

Interface coupling is represented by InterfaceConditions with two optional rows:

• scalar-like row (ScalarJump or RobinJump),
• flux-like row (FluxJump or RobinJump).

Typical forms:

\[\alpha_1 u_1 - \alpha_2 u_2 = g_s,\]

\[\beta_1 q_1 + \beta_2 q_2 = g_f,\]

where q_k are discrete normal-flux traces.

Outer Boundary Conditions

BorderConditions supports:

• Dirichlet(value),
• Neumann(value),
• Robin(α, β, value),
• Periodic().

Default side condition is homogeneous Neumann (0).

Embedded Interface Conditions

• Mono embedded condition: PenguinBCs.Robin(α, β, g).
• Diph embedded condition: InterfaceConditions(scalar=..., flux=...).

Important convention:

• Interface/jump coefficients and values are sampled at C_γ.
• This includes fixed and moving assembly paths.
• In moving slab assembly there is one natural slab C_γ sample set per step; temporal blending uses that same spatial sample set.

Callback Conventions

Diffusivity/source/interface coefficients can be:

• constants,
• space-dependent callbacks (x...),
• time-dependent callbacks (x..., t).

Examples:

D(x, y, t) = 1 + 0.2sin(2pi*t)
s(x, t) = x*(1 - x)
gγ(x, t) = exp(-t)

Geometry Assumptions

• Fixed models use a fixed cut-cell geometry (assembled_capacity).
• Moving models rebuild per-step slab geometry from SpaceTimeCartesianGrid.
• Halo and inactive rows are regularized to identity to keep the global linear system robust.

See Algorithms for discrete block systems and θ assembly.