Finite volume method

The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations.^[1] In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, 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 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.^[2]

Finite volume methods can be compared and contrasted with the finite difference methods, which approximate derivatives using nodal values, or finite element methods, 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.^[3]^[4]

^ LeVeque, Randall (2002). Finite Volume Methods for Hyperbolic Problems. ISBN 9780511791253.
^ Wanta, D.; Smolik, W. T.; Kryszyn, J.; Wróblewski, P.; Midura, M. (October 2021). "A Finite Volume Method using a Quadtree Non-Uniform Structured Mesh for Modeling in Electrical Capacitance Tomography". Proceedings of the National Academy of Sciences, India Section A: Physical Sciences. 92 (3): 443–452. doi:10.1007/s40010-021-00748-7.
^ Fallah, N. A.; Bailey, C.; Cross, M.; Taylor, G. A. (2000-06-01). "Comparison of finite element and finite volume methods application in geometrically nonlinear stress analysis". Applied Mathematical Modelling. 24 (7): 439–455. doi:10.1016/S0307-904X(99)00047-5. ISSN 0307-904X.
^ Ranganayakulu, C. (Chennu) (2 February 2018). "Chapter 3, Section 3.1". Compact heat exchangers : analysis, design and optimization using FEM and CFD approach. Seetharamu, K. N. Hoboken, NJ. ISBN 978-1-119-42435-2. OCLC 1006524487.{{cite book}}: CS1 maint: location missing publisher (link)

[1] LeVeque, Randall (2002). Finite Volume Methods for Hyperbolic Problems. ISBN 9780511791253.

[2] Wanta, D.; Smolik, W. T.; Kryszyn, J.; Wróblewski, P.; Midura, M. (October 2021). "A Finite Volume Method using a Quadtree Non-Uniform Structured Mesh for Modeling in Electrical Capacitance Tomography". Proceedings of the National Academy of Sciences, India Section A: Physical Sciences. 92 (3): 443–452. doi:10.1007/s40010-021-00748-7.

[3] Fallah, N. A.; Bailey, C.; Cross, M.; Taylor, G. A. (2000-06-01). "Comparison of finite element and finite volume methods application in geometrically nonlinear stress analysis". Applied Mathematical Modelling. 24 (7): 439–455. doi:10.1016/S0307-904X(99)00047-5. ISSN 0307-904X.

[4] Ranganayakulu, C. (Chennu) (2 February 2018). "Chapter 3, Section 3.1". Compact heat exchangers : analysis, design and optimization using FEM and CFD approach. Seetharamu, K. N. Hoboken, NJ. ISBN 978-1-119-42435-2. OCLC 1006524487.{{cite book}}: CS1 maint: location missing publisher (link)

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