Finite Volume Methods
Abstract
This chapter focuses on finite volume methods. The finite volume method is a discretization method that is well suited for the numerical simulation of various types (for instance, elliptic, parabolic, or hyperbolic) of conservation laws; it has been extensively used in several engineering fields, such as fluid mechanics, heat and mass transfer, or petroleum engineering. Some of the important features of the finite volume method are similar to those of the finite element method: it may be used on arbitrary geometries, using structured or unstructured meshes, and it leads to robust schemes. The finite volume method is locally conservative because it is based on a ``balance'' approach: a local balance is written on each discretization cell that is often called ``control volume;'' by the divergence formula, an integral formulation of the fluxes over the boundary of the control volume is then obtained. The fluxes on the boundary are discretized with respect to the discrete unknowns.
Related
- A potential exploring of a multi-dimensional coupled Burgers–Fisher–Huxley equation using finite volume and element methods (FVM/FEM)
- A structure preserving finite volume scheme for the multidimensional model in cardiac electrophysiology
- A Structure-Preserving and Thermodynamically Compatible Cell-Centered Lagrangian Finite Volume Scheme for Continuum Mechanics
- Small eigenvalues of negatively curved manifolds with finite volume
- Growth of Simple Closed Geodesics in Finite-Volume Hyperbolic Manifolds
BibTeX
@misc{eymard2026finite,
title = {{Finite Volume Methods}},
author = {Robert Eymard and Thierry Gallouët and Raphaèle Herbin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19748},
eprint = {2610.19748},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19748},
}
RIS
TY - UNPB
TI - Finite Volume Methods
AU - Robert Eymard
AU - Thierry Gallou{ë}t
AU - Rapha{è}le Herbin
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19748
UR - https://arxiv.org/abs/2610.19748
AB - This chapter focuses on finite volume methods. The finite volume method is a discretization method that is well suited for the numerical simulation of various types (for instance, elliptic, parabolic, or hyperbolic) of conservation laws; it has been extensively used in several engineering fields, such as fluid mechanics, heat and mass transfer, or petroleum engineering. Some of the important features of the finite volume method are similar to those of the finite element method: it may be used on arbitrary geometries, using structured or unstructured meshes, and it leads to robust schemes. The finite volume method is locally conservative because it is based on a ``balance'' approach: a local balance is written on each discretization cell that is often called ``control volume;'' by the divergence formula, an integral formulation of the fluxes over the boundary of the control volume is then obtained. The fluxes on the boundary are discretized with respect to the discrete unknowns.
ER -