V-HC-03 — finite slab, constant flux, adiabatic back

Tags: heat_conduction, heat_flux_bc, energy_balance

References:

• Carslaw & Jaeger (1959) §3.8 (constant-flux slab series)

Problem statement

Complements the semi-infinite cases at LONG times: the thermal wave reaches the back face — the regime cone-calorimeter simulations actually live in. Constant q″ at the top face, insulated back; exact solution T = T₀ + (q″L/k)·φ(x̃, Fo) with φ from exact/slab_series.jl (C&J §3.8), x̃ = z/L from the adiabatic back face.

Two parameter-free exact targets beyond the profiles:

• front–back plateau: ΔT → q″L/2k for Fo ≫ 1 (100 K here),
• global energy: mean temperature rises exactly at q″/(ρcL) (0.25 K/s here).

Quantities of interest (n = 320)

Comparison with other codes

The same case was solved with Gpyro 0.8200 at 201 nodes (Δz=0.1 mm); decks, outputs, and run provenance are committed under test/verification/reference/. Each code's error against the same exact solution is drawn below on a log scale, muted gray behind this solver's series — the signed linear-scale panel above shows where the error lives, this one compares magnitudes across codes.

Wall times at every ladder rung against the reference runs. Resolutions and simulated spans differ where noted (details in reference/timings.csv), so cross-code timings are indicative rather than a controlled benchmark; rungs at a matched resolution are directly comparable.

Solution overlays including the other codes' points: Tprofilesvs_codes, frontbackdThistoryvs_codes.

Convergence

Observed order 2 (L2), expected 2.0.

Solver configuration