##  [Finite-Volume Method](/finite-volume-method-0) 

 Definition

A discretization technique for partial differential equations that enforces local conservation by integrating the governing conservation laws over small control volumes and approximating fluxes across the faces, propagating cell-averaged quantities in time.

 

 

 

 

 

 





## Principle

Principle

Partition the domain into control volumes, integrate the conservation law over each control volume, convert divergence terms to surface fluxes via the divergence theorem, and approximate face fluxes to produce update formulas that conserve the quantity per cell.

 

 

 

 

 





## Demonstration

Demonstration

For 1D advection ∂u/∂t + ∂f(u)/∂x = 0, average u over cells, compute numerical fluxes f̂ at cell interfaces (e.g., upwind, Godunov flux), and update cell averages by net flux exchange ensuring discrete conservation.

 

 

 

 

## Misapplication

Misapplication

Using central-difference-like face fluxes without stabilization for strongly convective problems (high Peclet or Courant numbers) leading to nonphysical oscillations or violating monotonicity near discontinuities.

 

 

 

 

 





## Consequence

Consequence

Finite-volume methods yield locally conservative discretizations well-suited to conservation laws and compressible flows; with appropriate flux choices they capture shocks and discontinuities while preserving integral invariants.

 

 

 

 

## Reversal

Reversal

Finite-difference methods approximate differential operators at grid points and do not intrinsically enforce local integral conservation, making them less suited for discontinuous solutions unless modified.

 

 

 

 

 





## Boundary

Boundary

Applicable to PDEs expressible in divergence (conservation) form on domains that can be partitioned into control volumes; not the natural choice for problems better posed in global spectral bases or when pointwise smoothness is essential.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Closely related to finite-element methods (both partition domain), but finite-volume prioritizes local flux balance and conservation of integrated quantities while finite-element emphasizes variational consistency and basis function approximation.

 

 

 

 

 





## Synthesis

Synthesis

The finite-volume method is a control-volume discretization that enforces discrete local conservation by balancing fluxes across cell faces, producing robust schemes for conservation laws and fluid dynamics when combined with appropriate numerical fluxes.