Technical · numerical formulation

Spatial methods

Numerical methods are documented independently from programming language and calculation engine.

FDM — 1D

Current cell-centred finite-difference formulation, including physical closures, conservative flux construction, time-integration relationship and verification evidence.

technical preview

FEM — 1D

Current cell-centred P1 Galerkin formulation, including weak form, two-point Gauss assembly, row-sum mass lumping, half-cell completion and shared physical flux closures.

implemented / limited scope

FVM — 1D

The 1D finite-volume solver has an implemented-and-verified fixed-geometry/current-heating path. Geometry-changing welding stages remain unpromoted and retain a separate experimental-data boundary.

Numerical model at a glance

The equations behind the method pages

The full FDM and FEM pages carry the derivation, closures, assembly rules, stability references and verification boundaries. This compact layer makes the governing and semi-discrete objects visible before the reader enters the long-form documentation.

shared physical model

Flux-form thermal balance

\rho_\xi(T_\xi)c_{p,\xi}(T_\xi)\frac{\partial T_\xi}{\partial t} =\frac{\partial}{\partial x}\left(\kappa_\xi(T_\xi)\frac{\partial T_\xi}{\partial x}\right)+q'''_{v,\xi}

Material laws, sources and physical boundary/interface conditions are resolved before a spatial method is applied.

FDM · cell-centred

Conservative cell balance

\beta_i(T_i)\frac{dT_i}{dt} =\frac{\Phi_{i+1/2}-\Phi_{i-1/2}}{\Delta x_i}+Q_i(T,t)

A shared face-flux sequence makes the discrete conduction contribution auditable and keeps interface/boundary fluxes explicit.

FEM · P1 Galerkin

Lumped semi-discrete system

\mathbf M_L(\mathbf T)\dot{\mathbf T} =\mathbf f_v(\mathbf T,t)-\mathbf K_\kappa(\mathbf T)\mathbf T +\mathbf b_{\partial\Omega}(\mathbf T)+\mathbf f_{\Gamma}(\mathbf T,t)

Two-point quadrature, row-sum mass lumping and explicit half-cell completion expose how the cell-centred support closes the physical domain.

temporal contract

Explicit state advance

\mathbf T^{n+1}=\mathbf T^n+\Delta t\left(\frac{3}{2}\mathbf F^n-\frac{1}{2}\mathbf F^{n-1}\right)

Adams–Bashforth 2 is the current principal configuration; Forward Euler supplies the compatible starter and restart path.

Time integration remains a separate contract. The current principal explicit configuration is AB2 with a Forward-Euler starter/restart path; it is not part of the definition of FDM or FEM.