Finite Volume Methods

Finite volume methods (FVMs) are a class of numerical analysis methods used to solve partial differential equations (PDEs) numerically, much like the finite element method and finite difference methods. Finite volume methods work directly from the so-called strong form of the equation, whereas finite element methods are based on a variational formulation. The idea of the technique of control volumes (CVs) is to integrate the PDE on a certain set of control volumes to obtain discretized equations that conserve the value of every physical quantity on each volume. The FVM can be viewed as a special case combining both the finite difference method and the method of mean weighted residuals. The chapter demonstrates the FVM by applying it to examples of fluid flow problems, beginning with a scalar transport equation and then moving on to the Navier-Stokes equations.