Discretization¶

PyFVTool discretizes PDE terms into sparse matrix contributions that are assembled and solved by pyfvtool.solvePDE().

Available terms¶

Transient term¶

\[\alpha \frac{\partial \phi}{\partial t} \approx \alpha \frac{\phi^{n+1} - \phi^n}{\Delta t}\]

pf.transientTerm(c, dt, alpha)

Diffusion term¶

\[\nabla \cdot (\mathcal{D} \nabla \phi)\]

D_face = pf.geometricMean(D_cell)
pf.diffusionTerm(D_face)

Advection term¶

\[\nabla \cdot (\mathbf{u} \phi)\]

pf.convectionTerm(u_face)           # central difference
pf.convectionUpwindTerm(u_face)     # first-order upwind
pf.convectionTVDTerm(u_face, c, flux_limiter)  # TVD scheme

Source terms¶

pf.linearSourceTerm(beta_cell)      # β φ
pf.constantSourceTerm(gamma_cell)   # γ

Averaging methods¶

Cell-centred values must be interpolated to faces before use in face-based terms:

Function	Formula
`linearMean`	arithmetic average
`arithmeticMean`	arithmetic average
`geometricMean`	geometric average (recommended for diffusivity)
`harmonicMean`	harmonic average
`upwindMean`	upwind-biased

Assembling equations¶

Combine terms into a list and pass to solvePDE:

eqn = [pf.transientTerm(c, dt, 1.0),
       -pf.diffusionTerm(D_face),
       pf.constantSourceTerm(S_cell)]
pf.solvePDE(c, eqn)

The sign convention follows the PDE form: transient + advection + diffusion + source = 0, so diffusion typically appears with a negative sign.