Conservative by construction.
Checked by refinement.

The discrete balance

V_i is the cell volume. Φ is outward power through a face [W], including face area. Each internal face contributes equal and opposite powers to its two neighbours. Internal exchange therefore cancels when the balances are summed; the remaining energy residual is checked against boundary exchange and sources.

Temperature-dependent conduction

This same-material formula integrates conductivity along the temperature difference (the Kirchhoff transformation). At a material interface, the implementation solves for a common face temperature that gives equal conductive powers through the two half-cells. It assumes positive scalar conductivity and perfect contact. Related conductivity-treatment literature is distinguished from our implementation.

Time and nonlinear solution

Backward Euler uses the new-time fluxes and enthalpy change. A damped Newton iteration solves the coupled residual with a sparse linear solve. Failed nonlinear steps can be retried with a smaller time step. This retry mechanism controls nonlinear convergence; it is not an estimate of temporal truncation error.

Solving the algebraic system tightly does not remove mesh error. Spatial resolution, time step and phase regularization must be assessed independently for the intended observable.