Numerical Algorithms & Routines

This page summarizes the high-level routines implemented by the package.

1. Fixed steady assembly (assemble_steady_mono!, assemble_steady_diph!)

• Build the diffusion system with PenguinDiffusion.assemble_steady_*!.
• Build advection operators at time t.
• Add transport blocks (ω/γ per phase).
• Apply transport box BC rows on ω rows.
• Re-apply row-identity constraints for inactive/halo rows.

1. Fixed unsteady assembly (assemble_unsteady_mono!, assemble_unsteady_diph!)

• Assemble at t + θΔt.
• For θ != 1, scale ω rows by θ and add -(1-θ)A*uⁿ correction to RHS.
• Add mass matrix on each ω block: M = V / Δt and bω += M .* uωⁿ.
• Re-apply activity constraints and clear solver cache.

1. Moving unsteady assembly (assemble_unsteady_mono_moving!, assemble_unsteady_diph_moving!)

• Delegate slab geometry + moving diffusion/mass terms to PenguinDiffusion.assemble_unsteady_*_moving!.
• Evaluate advection at τ = t + θΔt.
• Build moving advection operators on slab-reduced capacities.
• Use relative interface velocity on γ: uγ_rel = uγ - wγ.
• Append advection blocks, apply transport BCs, enforce row identities.

1. Time marching (solve_unsteady!, solve_unsteady_moving!)

• Accepts u0 as either phase ω blocks or full system vector.
• Advances with variable last step (min(dt, tend-t)).
• Solves each step with PenguinSolverCore.solve!.
• Fixed path can reuse constant operators when data are time-independent.
• Moving path reassembles every step.

1. Geometry updates (rebuild!)

• Rebuilds capacities (CartesianOperators.rebuild!) from new moments.
• Reconstructs diffusion operators/model and periodic flags.
• Refreshes advection operators for requested time.