Editing Finite volume method (section)

{{Short description|Method for representing and evaluating partial differential equations}}
{{Differential equations}}

The '''finite volume method''' ('''FVM''') is a method for representing and evaluating [[partial differential equation]]s in the form of algebraic equations.<ref>{{cite book|last=LeVeque|first=Randall|date=2002|title=Finite Volume Methods for Hyperbolic Problems|url=https://www.cambridge.org/core/books/finite-volume-methods-for-hyperbolic-problems/97D5D1ACB1926DA1D4D52EAD6909E2B9|isbn=9780511791253}}</ref>
In the finite volume method, volume integrals in a partial differential equation that contain a [[divergence]] term are converted to [[surface integral]]s, using the [[divergence theorem]]. 
These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are [[Conservation law (physics)|conservative]]. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many [[computational fluid dynamics]] packages.
"Finite volume" refers to the small volume surrounding each node point on a mesh.<ref>{{Cite journal |last1=Wanta |first1=D. |last2=Smolik |first2=W. T. |last3=Kryszyn |first3=J. |last4=Wróblewski |first4=P. |last5=Midura |first5=M. |date=October 2021 |title=A Finite Volume Method using a Quadtree Non-Uniform Structured Mesh for Modeling in Electrical Capacitance Tomography |journal=Proceedings of the National Academy of Sciences, India Section A: Physical Sciences |volume=92 |issue=3 |pages=443–452 |language=en|doi=10.1007/s40010-021-00748-7 |doi-access=free}}</ref>

Finite volume methods can be compared and contrasted with the [[finite difference method]]s, which approximate derivatives using nodal values, or [[finite element method]]s, which create local approximations of a solution using local data, and construct a global approximation by stitching them together. In contrast a finite volume method evaluates exact expressions for the ''average'' value of the solution over some volume, and uses this data to construct approximations of the solution within cells.<ref>{{Cite journal|last1=Fallah|first1=N. A.|last2=Bailey|first2=C.|last3=Cross|first3=M.|last4=Taylor|first4=G. A.|date=2000-06-01|title=Comparison of finite element and finite volume methods application in geometrically nonlinear stress analysis|journal=Applied Mathematical Modelling|language=en|volume=24|issue=7|pages=439–455|doi=10.1016/S0307-904X(99)00047-5|issn=0307-904X|doi-access=free}}</ref><ref>{{Cite book|last=Ranganayakulu, C. (Chennu)|title=Compact heat exchangers : analysis, design and optimization using FEM and CFD approach|others=Seetharamu, K. N.|isbn=978-1-119-42435-2|location=Hoboken, NJ|chapter=Chapter 3, Section 3.1|date=2 February 2018 |oclc=1006524487}}</ref>