Finite Volume Methods and their Analysis

Abstract

Finite volume methods are widely used and highly successful in computing solutions to conservation laws, such as those occurring in fluid dynamics: but little analysis of their behaviour has been carried out. In this paper we use model problems in one and two dimensions to initiate a study of such methods, especially the cell vertex method. In one dimension it shows that finite volume methods give accurate flux values at volume boundaries: thus, even for the self-adjoint equation (aø′=f, these gradient values can be more accurate than for finite element methods. The cell vertexes schemes are aimed at convection dominated problems and have highly advantageous properties for convection-diffusion problems: but we give error bounds for these four-point approximations to the pure diffusion problem in one dimension which indicates their remarkable robustness. Finally, an analysis of the cell vertex scheme for the pure advection problem in two dimensions is given. It explains the insensitivity of the method to mesh stretching in the coordinate directions, maintaining second-order accuracy on any mesh.