Finite volumes numerical schemes

Finite volume methods are in some ways complementary to finite difference methods [AF 501] and finite element methods [AF 503][AF 504][AF 505] . The data structure is very similar to that of finite differences when these methods are used on a Cartesian mesh, while allowing greater geometric flexibility on non-Cartesian meshes, as is the case for finite element methods. Finite volume methods are also widely used for the numerical discretization of nonlinear partial differential equations, such as the equations of compressible gas dynamics. They are also very robust methods. These properties explain their appeal. However, the construction principle, which relies on integral rather than differential or weak formulas, is different from finite difference or finite element methods.

The aim of this dossier is to present, as simply as possible, a few rules for constructing various finite volume schemes. The more technical aspects of convergence proofs are not covered.