The Finite Volume Method

Abstract

Similar to other numerical methods developed for the simulation of fluid flow, the finite volume method transforms the set of partial differential equations into a system of linear algebraic equations. Nevertheless, the discretization procedure used in the finite volume method is distinctive and involves two basic steps. In the first step, the partial differential equations are integrated and transformed into balance equations over an element. This involves changing the surface and volume integrals into discrete algebraic relations over elements and their surfaces using an integration quadrature of a specified order of accuracy. The result is a set of semi-discretized equations. In the second step, interpolation profiles are chosen to approximate the variation of the variables within the element and relate the surface values of the variables to their cell values and thus transform the algebraic relations into algebraic equations. The current chapter details the first discretization step and presents a broad review of numerical issues pertaining to the finite volume method. This provides a solid foundation on which to expand in the coming chapters where the focus will be on the discretization of the various parts of the general conservation equation. In both steps, the selected approximations affect the accuracy and robustness of the resulting numerics. It is therefore important to define some guiding principles for informing the selection process.