15 - Finite Volume and Finite Element Methods

Summary

Introduction

Except for Chapter 14 that discussed spectral methods, all earlier chapters have discussed the method of finite difference that still remains a popular method for solving partial differential equations. While employing the finite difference method, the differential form of the conservation law of fluid flow was utilized and the partial derivatives present in the differential form of the conservation equations were approximated using the appropriate finite difference expressions. In this chapter, the finite volume method will be introduced and applied to solve fluid flow problems. The basis of the finite volume method is the utilization of the conservation law of fluid flow in the integral form.

Integral Form of Conservation Law

The physical (“fluid” in this case) system is governed by a set of conservation laws that include conservation of mass, conservation of momentum, and conservation of energy. These conservation laws are mostly written in differential form. The method of finite difference requires the differential form of the conservation laws, whereas the finite volume method entails the integral form of the conservation laws for a fixed physical domain. Assume the existence of a physical domain, Ω, with the boundary of the domain, indicated by Ω. Then, the canonical conservation equation assuming that the physical domain is fixed is of the following form:

where U is the conserved state, is the flux of the conserved state, is the outward directed unit normal on the boundary of the domain, and S denotes a source term. By applying the Gauss divergence theorem, the aforementioned conservation law can be written as a partial differential equation. The Gauss divergence theorem states that

Equation (15.1) then becomes

Applying the Leibnitz rule for differentiation under the integral sign and combining the volume integrals, one obtains

As Equation (15.4) must be valid for any arbitrary domain, Ω, the integrand that appears in Equation (15.4) must be zero everywhere, or equivalently,

Equation (15.5) specifies the conservation law in the differential form as a partial differential equation.