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Home/Math Visualization/Finite-Volume Advection-Diffusion 2D

Finite-Volume Advection-Diffusion 2D

Conservative scalar transport on a 2D grid: face fluxes, upwind vs central interpolation, Peclet number, CFL, and numerical diffusion.

Finite-volume scheme

1
15°
0.015
3

Measured values

Cell Peclet number1.149
CFL number0.263
Time step Δt0.00396
Grid58×40
Scheme hintbounded

The demo uses periodic boundaries and an explicit time step chosen from simple CFL/diffusion limits. Central differencing can show wiggles at high Peclet number, while upwind is more bounded but more diffusive.

Live graphs

About this model

Finite-volume methods update cell averages by balancing fluxes through the faces of each control volume. This simulator evolves a scalar advection-diffusion equation on a periodic 2D grid, computing east/west/north/south face fluxes and comparing upwind versus central interpolation. Upwind fluxes are more dissipative but bounded; central differencing is sharper but can oscillate when the cell Peclet number is high. The canvas shows the transported scalar field, a velocity vector, sample face fluxes, CFL, time step, and Peclet number.

Who it's for: Scientific computing, finite-volume CFD, numerical PDEs, transport phenomena, and applied math courses.

Key terms

  • Finite volume method
  • Advection-diffusion
  • Face flux
  • Upwind scheme
  • Central differencing
  • Peclet number

How it works

Finite-volume advection-diffusion simulator: conservative face fluxes, upwind versus central interpolation, Peclet number, and scalar transport on a 2D grid.

Key equations

∂φ/∂t + ∇·(uφ) = ∇·(Γ∇φ), integrated over each control volume
Face flux: F_e φ_e − Γ(φ_E−φ_P)/Δx; Pe = |u|Δx/Γ

Frequently asked questions

Why do finite-volume methods focus on face fluxes?
The integral conservation law says that a cell average changes because material crosses its faces. Computing numerical face fluxes makes conservation local and explicit.
Why can central differencing oscillate?
When advection dominates diffusion, the cell Peclet number is high and central interpolation lacks enough numerical damping. Upwind uses the upstream value, which stabilizes the transport at the cost of extra diffusion.