Finite volume method

The finite volume method is a numerical method for solving conservation laws by balancing fluxes across the boundaries of discrete control volumes.

Control volume: A finite region over which a conservation law is integrated to track the change in a quantity. Each discrete cell is treated as a control volume whose net boundary flux sets its change.
Divergence theorem: A theorem equating the integral of a field’s divergence over a region with its outward flux across the boundary. It converts the conservation law’s volume integral into the boundary flux sum.
Computational fluid dynamics: The numerical study of fluid flows using discretized governing equations. Finite volume schemes are a standard approach for simulating fluid conservation laws.
Finite difference method: A numerical method that approximates derivatives using values at discrete points. It discretizes differential expressions directly rather than beginning with cell-integrated balances.
Numerical flux: An approximate flux shared by neighboring cells at a discrete interface. A single shared face flux makes the exchange between adjacent volumes cancel exactly.
Partial differential equation: An equation relating a function of several variables to its partial derivatives. Finite volume schemes approximate partial differential equations, especially in conservation form.
Euler equations: The nonlinear conservation equations governing inviscid fluid motion. Their shock-containing solutions are often computed with finite volume flux schemes.
Finite element method: A numerical method that approximates solutions with basis functions over mesh elements. Its variational formulation differs from the direct flux accounting typical of finite volumes.
Finite-volume discretization: The conversion of an integral conservation law into algebraic balances over a mesh of cells. This step turns the continuous balance law into the method’s discrete equations.
Unstructured grid: A mesh whose cells need not follow a regular, repeating connectivity pattern. Finite volume balances apply to irregular cell shapes and neighbor connections.

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