Advanced Numerical and Semi-Analytical Methods for Differential Equations

7Finite Volume Method

7.1 Introduction

Finite volume method (FVM) is another widely used numerical technique [1,2]. The fundamental conservation property of the FVM makes it the preferred method compared to various existing methods viz. finite difference method (FDM), finite element method (FEM), etc. In this approach, similar to the known numerical methods like FDM (Chapter 5) or FEM (Chapter 6), the volumes (elements or cells) are evaluated at discrete places over a meshed geometry. Then, the involved volume integrals of respective differential equation containing divergence term are converted to surface integrals by using well‐known divergence theorem [3,4]. Further, the simulated differential equation gets transformed over differential volumes into discrete system of algebraic equations. In the final step, the system of algebraic equations is solved by standard methods to determine the dependent variables.

Aforementioned FVM has various other advantages in handling differential equations occurring in science and engineering problems. For instance, a key feature of the FVM is formulation of physical space (domain of the differential equation) on unregulated polygonal meshes. Another feature is that the FVM is quite easy to implement various boundary conditions in a noninvasive manner, because the involved unknown variables are calculated at the centroids of the volume elements, rather than at their boundary faces [5]. These characteristics of the FVM have made it suitable ...