The Finite Volume Method
A discretization that enforces conservation exactly by tracking fluxes across the faces of control volumes.
Conservation built in
The finite volume method divides the domain into control volumes (cells) and tracks the average of the conserved quantity in each. It integrates the governing conservation law over each cell, turning volume integrals of divergences into surface integrals of fluxes across cell faces. Because the flux leaving one cell exactly equals the flux entering its neighbor, the total conserved quantity is preserved to machine precision. This discrete conservation is the method's defining strength for problems governed by conservation laws.
Fluxes at the faces
The core task is computing the flux at each face from the cell-average values on either side. For smooth flows a centered average suffices, but for advection-dominated and discontinuous flows an upwind or Riemann-solver-based flux is needed to respect the direction of information flow and to keep the scheme stable. The Godunov method solves a local Riemann problem at each face; approximate Riemann solvers (Roe, HLL, HLLC) trade some accuracy for speed.
Achieving higher order
- First-order: use cell averages directly; robust but diffusive
- Higher-order: reconstruct a polynomial within each cell (MUSCL, WENO) before computing fluxes
- Slope limiters keep the reconstruction from creating new oscillations near discontinuities
Why it suits fusion and fluid problems
The finite volume method handles unstructured and irregular meshes naturally, since it needs only cell volumes and face areas, not a structured coordinate grid. Combined with its exact conservation, this makes it the standard for compressible fluid dynamics and for the fluid (MHD) description of plasmas, where conserving mass, momentum, energy, and magnetic flux is physically essential. It pairs with WENO reconstruction for high-order shock capturing and with the method of lines for time integration.