Meshing, Geometric Modeling and Numerical Simulation 3

Chapter 7Meshes and Finite Volume Calculation

Among the classic methods for solving problems formulated in partial differential equations and along with finite element methods (previous chapter), we find finite volume methods, which, as we shall see, allow hyperbolic-type equations to be addressed.

In this chapter, we want to show how the methodologies seen in this book make it possible to efficiently build the data structures of solvers based on the finite volume method in two dimensions when considering unstructured simplicial meshes. Finite volume methods have been proposed as an alternative to the finite element method to solve a particular type of equation: hyperbolic problems that have an unsymmetrical structure.

In order to understand the formulation of a finite volume solver, we briefly present the foundations of the finite volume method in one dimension with a fundamental hyperbolic equation: the advection equation. Then we briefly describe the extension to systems of hyperbolic equations with the case of the Euler equation in two dimensions. The algorithms for constructing the data structures necessary for the implementation of this type of method will be specified. In the last section, we show some numerical results obtained with this approach. This introduction is obviously non-exhaustive and we refer the reader to the works of [Hirsch-1988], [Hirsch-1990] or [Toro-2009] for a full review of this type of methods.

7.1. Presentation of the finite volume method with ...