Numerical Solution of the Navier-Stokes Equations by the Finite Volume Method

In this series of articles devoted to the numerical solution of Navier-Stokes equations, we present several methods of resolution based on different types of discretization methods for partial differential equations. We have already presented :

• on the one hand, the finite-difference method, where derivatives are replaced by differential quotients [AF 1 404] ; this method corresponds to the expression of a balance of the quantities represented by the physical model at each point of the mesh;

• on the other hand, the finite element method [AF 1 407] where, after having given an equivalent formulation of the problem via Green's formula in an appropriate space of test functions, which corresponds roughly to an extension of the generalized integration by parts formula (or, more generally, to the use of derivation in the sense of distributions) and leads to the application of the principle of virtual work, the solution is decomposed into a finite basis that is well adapted numerically; this is equivalent to projecting the exact solution of an infinite-dimensional space onto a finite-dimensional space [AF 503][AF 504][AF 505] ; this finite-element method has the advantage of being able to solve the Navier-Stokes equation on unstructured meshes, and is also well suited when the domain Ω is of any shape with a curved boundary ∂Ω ;

• there are other discretization methods, such as the variational finite-difference...