Finite Volume Methods: Foundation and Analysis
Authors: Barth Timothy , Herbin Raphaèle , Ohlberger Mario 
Published: December 2017

Finite volume methods are a class of discretization schemes resulting from the decomposition of a problem domain into nonoverlapping control volumes. Degrees of freedom are assigned to each control volume that determine local approximation spaces and quadratures used in the calculation of control volume surface fluxes and interior integrals. An imposition of conservation and balance law statements in each and every control volume constrains surface fluxes and results in a coupled system of equations for the unknown degrees of freedom that must be solved by a numerical method.

Finite volume methods have proved highly successful in approximating the solution to a wide variety of conservation and balance laws. They are extensively used in fluid mechanics, meteorology, electromagnetics, semiconductor device simulation, materials modeling, heat transfer, models of biological processes, and many other engineering problems governed by conservation and balance laws that may be written in integral control volume form.

This chapter reviews elements of the foundation and analysis of modern finite volume methods for approximating hyperbolic, elliptic, and parabolic partial differential equations. These different equations have markedly different continuous problem regularity and function spaces (e.g., , , and ) that must be adequately represented in finite‐dimensional discretizations. Particular attention is given to finite volume discretizations yielding numerical solutions that inherit properties of the underlying continuous solutions such as maximum (minimum) principles, total variation control, stability, global entropy decay, and local balance law conservation while also having favorable accuracy and convergence properties on structured and unstructured meshes.

As a starting point, a review of scalar nonlinear hyperbolic conservation laws and the development of high‐order accurate schemes for discretizing them is presented. A key tool in the design and analysis of finite volume schemes suitable for discontinuity capturing is discrete maximum principle analysis. A number of mathematical and algorithmic developments used in the construction of numerical schemes possessing local discrete maximum principles are reviewed in one and several space dimensions. These developments include monotone fluxes, TVD discretization, positive coefficient discretization, nonoscillatory reconstruction, slope limiters, strong stability preserving time integrators, and so on. When available, theoretical results concerninga priorianda posteriorierror estimates and convergence to entropy weak solutions are given.

A review of the discretization of elliptic and parabolic problems is then presented. The tools needed for the theoretical analysis of the two point flux approximation scheme for the convection diffusion equation are described. Such schemes require an orthogonality condition on the mesh in order for the numerical fluxes to be consistent. Under this condition, the scheme may be shown to be monotone. A weak formulation of the scheme is derived, which facilitates obtaining stability, convergence, and error estimate results. The discretization of anisotropic problems is then considered and a review is given of some of the numerous schemes that have been designed in recent years, along with their properties. Parabolic problems are then addressed, both in the linear and nonlinear cases.

A discussion of further advanced topics is then given including the extension of the finite volume method to systems of hyperbolic conservation laws. Numerical flux functions based on an exact or approximate solution of the Riemann problem of gas dynamics are discussed. This is followed by the review of another class of numerical flux functions for symmetrizable systems of conservation laws that yield finite volume solutions with provable global decay of the total mathematical entropy for a closed entropy system, often referred to as entropy stability.

Finally, a detailed review of the discretization of the steady‐state incompressible Navier–Stokes equations using the Marker‐And‐Cell (MAC) finite volume method is then presented. The MAC scheme uses a staggered mesh discretization for pressure and velocities on primal and dual control volumes. After reformulating the MAC scheme in weak form, analysis results concerning stability, weak consistency, and convergence are given.