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| Finite Volume Method | |
|---|---|
| Name | Finite Volume Method |
| Caption | Conservation cell example |
Finite Volume Method
The Finite Volume Method is a numerical technique for solving partial differential equations that emphasizes local conservation on control volumes. It is widely used in computational fluid dynamics and engineering fields and interfaces with software and institutions that advance numerical simulation.
The Finite Volume Method has development ties with pioneers and institutions such as John von Neumann, Courant Institute, Los Alamos National Laboratory, École Polytechnique, Imperial College London, and Massachusetts Institute of Technology, and it is applied in contexts related to NASA, European Space Agency, National Aeronautics and Space Administration, Brookhaven National Laboratory, and Lawrence Livermore National Laboratory. Influential figures and works like Hermann Weyl, Richard Courant, André-Marie Ampère, Leonhard Euler, and Carl Friedrich Gauss underpin the mathematical foundations, while software and projects from OpenFOAM Foundation, ANSYS, Siemens, Microsoft Research, and IBM Research implement practical solvers. The method is connected to numerical analysis advances from Alan Turing, John von Neumann, Kurt Gödel, Norbert Wiener, and Stephen Hawking and has been disseminated through venues such as International Congress of Mathematicians, SIAM, IEEE, Royal Society, and American Mathematical Society.
The mathematical formulation builds on conservation laws framed by contributors including Isaac Newton, James Clerk Maxwell, Joseph Fourier, Sadi Carnot, and Daniel Bernoulli. Governing equations like the Navier–Stokes equations trace conceptual lineage through Claude-Louis Navier and George Gabriel Stokes, and their discrete analogs rely on divergence theorems associated with Carl Friedrich Gauss and integral formulations linked to Pierre-Simon Laplace. Variational and weak formulations draw on work from David Hilbert, Emmy Noether, Bernhard Riemann, Siméon Denis Poisson, and Augustin-Louis Cauchy, and matrix and linear algebra tools involve contributions from Gustav Kirchhoff, Arthur Cayley, James Joseph Sylvester, and Évariste Galois.
Discretization strategies reference grids and meshes used across projects at CERN, Brookhaven National Laboratory, Oak Ridge National Laboratory, National Institute of Standards and Technology, and Lawrence Berkeley National Laboratory. Structured and unstructured meshes connect to mesh-generation advances at Carnegie Mellon University, Stanford University, California Institute of Technology, Princeton University, and University of Cambridge. Techniques such as cell-centered, vertex-centered, and staggered arrangements are used in tools developed by Fermilab, SLAC National Accelerator Laboratory, Bell Labs, and NASA Ames Research Center. Mesh adaptation and refinement concepts were advanced in collaborations involving DARPA, European Commission, United Nations Educational, Scientific and Cultural Organization, Max Planck Society, and CNRS.
Flux computation methods build on numerical flux and Riemann solvers associated with researchers from University of Oxford, University of Chicago, Yale University, Columbia University, and University of California, Berkeley. Upwind, Godunov, and Roe-type schemes connect to names and places like Sergei Novikov, Peter Lax, Lars Hörmander, John von Neumann School, and research centers such as Los Alamos National Laboratory and Sandia National Laboratories. High-resolution schemes, total variation diminishing ideas, and limiter functions are developed and promoted through conferences at SIAM, ACM, IEEE, Royal Society of London, and European Mathematical Society.
Specification of boundary and initial conditions is informed by standards and practices at institutions such as International Organization for Standardization, American Institute of Aeronautics and Astronautics, European Space Agency, NASA Jet Propulsion Laboratory, and United States Geological Survey. Treatments including Dirichlet, Neumann, and Robin-like conditions in practical implementations draw on mathematical frameworks from Augustin-Louis Cauchy, Siméon Denis Poisson, Joseph-Louis Lagrange, and applied work at MIT Lincoln Laboratory, Johns Hopkins University, Harvard University, and Yale University.
Error estimation and convergence theories relate to foundational results from Andrey Kolmogorov, Norbert Wiener, Stefan Banach, Henri Lebesgue, and Emil Artin. Stability and consistency criteria reflect developments influenced by Richard Bellman, Kurt Friedrichs, James H. Wilkinson, and numerical linear algebra progress from John von Neumann and Alan Turing. Adaptive error control methodologies have been refined in programs at European Research Council, National Science Foundation, Wellcome Trust, and research groups at University of Cambridge and University of Oxford.
Applications span aerospace and turbine design in projects at Boeing, Airbus, Rolls-Royce plc, General Electric, and Pratt & Whitney; climate and weather modeling in institutions like National Oceanic and Atmospheric Administration, Met Office, European Centre for Medium-Range Weather Forecasts, National Weather Service, and Intergovernmental Panel on Climate Change; and astrophysical simulations at European Southern Observatory, Space Telescope Science Institute, Max Planck Institute for Astrophysics, Royal Astronomical Society, and Jet Propulsion Laboratory. Industrial and biomedical examples are implemented by Siemens Healthineers, Medtronic, Pfizer, GlaxoSmithKline, and Bayer. Case studies and benchmarks originate from collaborative programs at Sandia National Laboratories, Los Alamos National Laboratory, Lawrence Berkeley National Laboratory, Princeton Plasma Physics Laboratory, and CERN.
Category:Numerical methods