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The finite volume method follows the amount of a conserved quantity inside each control volume. Its unknowns are cell averages, rather than values at isolated points: a cell's width times its stored average approximates that cell's total. A numerical flux transfers an amount across a face. Using one shared transfer for both neighboring cells makes the loss from one exactly the gain of the other, as explained in shared face flux cancellation.^[ref]

For a one-dimensional scalar conservation law without sources, let $u_t+{\partial }_{x}f(u)=0$, with conservation law flux $f$. Fix cells $I_j=[x_{j-1/2},x_{j+1/2}]$ of widths $h_j>0$. Define the exact cell average and its numerical approximation byThe integral balance over one cell and one time interval of length $k>0$ isFor a smooth solution this follows by integration of the differential equation; with jump discontinuities, it expresses the conservation law in integral form, using appropriate face traces when they exist. Replacing each time-averaged face conservation law flux by a numerical flux givesThus the finite volume method implements a balance directly, without needing a pointwise derivative at a jump. The tube balance in Riemann Problems and Jupyter Solutions by David I. Ketcheson, Randall J. LeVeque, and Mauricio J. del Razo (Advection, §Conservation of mass) motivates this construction from conservation of mass.

This is useful for shock waves: their speed is constrained by the Rankine-Hugoniot condition, itself an integral balance. A nonconservative approximation can give a visibly plausible jump with the wrong speed. Conservation is one requirement; a nonlinear scalar conservation law also needs the correct entropy solution, and the chosen numerical flux and update need their own stability and accuracy analysis.^[ref] The original periodic upwind finite volume advection example isolates the shared-transfer mechanism in a problem with a known translated solution.

A numerical flux estimates the rate of transfer across a cell face from nearby states. In the finite volume method update, $F_{j+1/2}^n$ approximates the conservation law flux averaged over the time step; $k{F}_{j+1/2}^n$ is the transferred amount. For a two-state numerical flux $F(L,R)$ approximating a scalar conservation law with conservation law flux $f$, consistency of a numerical method requires $F(U,U)=f(U)$. Thus equal states recover the physical transfer rate. With constant equal-width cells, this also preserves a constant state because its incoming and outgoing numerical fluxes agree.^[ref]

Consistency and shared face flux cancellation answer different questions: matching $f$ at equal states approximates the intended equation, while sharing one oriented face value preserves the total. The example periodic upwind finite volume advection uses $F(L,R)=aL$ for $a>0$, so the arriving state comes from the left. The Godunov numerical flux uses an entropy solution of a local Riemann problem; the Engquist-Osher flux provides a different construction for a scalar conservation law. A monotone conservative scheme uses a numerical flux nondecreasing in its left state and nonincreasing in its right state, together with a time step that makes the complete update monotone. Neither two-state consistency nor conservation by itself supplies that time-step restriction or a general stability or order of a numerical method guarantee.