The Finite Volume Method
Finite volumes integrate conservation laws over control cells so that fluxes between neighbors cancel exactly, enforcing conservation by construction.
Conservation as the organizing principle
The finite volume method (FVM) starts from the integral form of a conservation law. The domain is partitioned into control volumes, and the equation is integrated over each cell. By the divergence theorem, the volume integral of a flux divergence becomes a surface integral over cell faces. The discrete unknown is the cell average, and the scheme tracks how face fluxes move that average in time.
Because a flux leaving one cell across a shared face is exactly the flux entering its neighbor, the total conserved quantity changes only through the domain boundary. This telescoping property makes FVM the natural choice for mass, momentum, and energy transport where discrete conservation is non-negotiable.
Reconstruction and flux evaluation
The core modeling choice is how to compute the face flux from cell averages. A reconstruction step estimates the state at each face from surrounding cell values; a numerical flux function then combines the left and right face states into a single flux. For hyperbolic systems this flux typically comes from an approximate Riemann solver, which respects the direction of wave propagation and introduces controlled upwind dissipation.
Accuracy and geometry
First-order FVM uses piecewise-constant reconstruction and is robust but diffusive. Higher order comes from linear or polynomial reconstruction with slope limiting to suppress spurious oscillations near discontinuities. Unlike finite differences, FVM works naturally on unstructured polygonal and polyhedral meshes, so it handles complex geometry without coordinate transformations.
- Cell-centered vs vertex-centered layouts of the unknowns
- Numerical flux functions: upwind, Lax-Friedrichs, Roe, HLL/HLLC
- Slope limiters preserving monotonicity at second order
- Discrete conservation guaranteed regardless of mesh quality
FVM dominates computational fluid dynamics and neutral-gas and edge-plasma transport modeling, where conserving particle and energy budgets across long simulations is essential for physical credibility.