Finite volume methods. (English) Zbl 0843.65070

Morton, K. W.

Griffiths, D. F. (ed.) et al., Numerical analysis 1995. Proceedings of the 16th Dundee conference on numerical analysis, University of Dundee, UK, June 27-30, 1995. Harlow: Longman. Pitman Res. Notes Math. Ser. 344, 123-139 (1996).

Summary: To the numerical analyst, finite volume methods are low order, nonconforming Petrov-Galerkin methods of no particular significance. But to the aeronautical engineer they are key design tools, well suited to the efficient modelling of nonlinear conservation laws in complicated three-dimensional geometries. In view of their dominance in this area of computational fluid dynamics, and because of the need to resolve a number of important issues and make improvements in key areas, these methods merit and require more attention from numerical analysts.
In this lecture, I briefly review the range of methods on structured and unstructured meshes that are in use, and their links to more familiar finite difference and finite element methods. This is followed by a selection from the numerical analysis results that have been established, particularly in regard to their robustness for singular perturbation problems. Finally, I outline some of the outstanding issues and situations for which more analysis is needed. In each case the cell vertex scheme is used as a key exemplar of the methods.
For the entire collection see [Zbl 0837.00017].

MSC:

65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76M10	Finite element methods applied to problems in fluid mechanics
35L65	Hyperbolic conservation laws
35B25	Singular perturbations in context of PDEs

Keywords:

comparison of methods; finite element methods; finite volume methods; Petrov-Galerkin methods; nonlinear conservation laws; computational fluid dynamics; singular perturbation problems; cell vertex scheme