# OpenFOAM: Finite Volume Method

The finite-volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999].

## 1 Introduction

The purpose of any discretisation practice is to transform one or more partial differential equations into a corresponding system of algebraic equations. The solution of this system produces a set of values which correspond to the solution of the original equations at some pre-determined locations in space and time, provided certain conditions, to be defined later, are satisfied. The discretisation process can be divided into two steps: the discretisation of the solution domain and equation discretisation (Hirsch , Muzaferija ).

The discretisation of the solution domain produces a numerical description of the computational domain, including the positions of points in which the solution is sought and the description of the boundary. The space is divided into a finite number of discrete regions, called control volumes or cells. For transient simulations, the time interval is also split into a finite number of time-steps. Equation discretisation gives an appropriate transformation of terms of governing equations into algebraic expressions.

This Chapter presents the Finite Volume method (FVM) of discretisation, with the following properties:

• The method is based on discretising the integral form of governing equations over each control volume. The basic quantities, such as mass and momentum, will therefore be conserved at the discrete level.
• Equations are solved in a fixed Cartesian coordinate system on the mesh that does not change in time. The method is applicable to both steady-state and transient calculations.
• The control volumes can be of a general polyhedral shape, with a variable number of neighbours, thus creating an arbitrarily unstructured mesh. All dependent variables share the same control volumes, which is usually called the colocated or non-staggered variable arrangement (Rhie and Chow , Peri´c ).
• Systems of partial differential equations are treated in the segregated way (Patankar and Spalding , van Doormaal and Raithby ), meaning that they are solved one at a time, with the inter-equation coupling treated in the explicit manner. Non-linear differential equations are linearised before the discretisation and the non-linear terms are lagged.

## 2 Discretisation of the Solution Domain

Discretisation of the solution domain produces a computational mesh on which the governing equations are subsequently solved. It also determines the positions of points in space and time where the solution is sought. The procedure can be split into two parts: discretisation of time and space.