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Computational Methods for Fluid Dynamics pp 81–110Cite as

Finite Volume Methods

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Abstract

The finite volume (FV) method is described for the generic scalar transport equation in this chapter, including the approximation of surface and volume integrals and the use of interpolation to obtain variable values and derivatives at locations other than cell centers. Development of higher-order schemes and simplification of the resulting algebraic equations using the deferred-correction approach is also described. Special attention is paid to the analysis of discretization errors caused by interpolation and integral approximations. Finally, implementation of the various boundary conditions is discussed. The chapter closes with application of some of the basic methods to several examples using Cartesian grids.

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Notes

  1. 1.

    In this section we are looking at schemes to define \(\phi \) and/or its derivative. In Sect. 11.3, we will examine so-called flux-corrected transport (FCT).

  2. 2.

    N.B.: This is numerical dispersion; some cases, e.g., nonlinear processes and waves in the ocean, may exhibit real physical dispersion.

  3. 3.

    The Spalding hybrid scheme (Sect. 4.4.5) can have this effect as well, as seen in Freitas et al. (1985), where the replacement of that scheme by QUICK (Sect. 4.4.3) in a simulation with a three-dimensional unsteady code revealed vortexes and other three-dimensional effects that had been hidden by the spurious numerical diffusion of the hybrid scheme.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Stanford University, Stanford, CA, USA

    Joel H. Ferziger

  2. University of Duisburg-Essen, Duisburg, Germany

    Milovan Perić

  3. Department of Civil and Environmental Engineering, Stanford University, Stanford, CA, USA

    Robert L. Street

Authors
  1. Joel H. Ferziger
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  2. Milovan Perić
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  3. Robert L. Street
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Correspondence to Milovan Perić .

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Ferziger, J.H., Perić, M., Street, R.L. (2020). Finite Volume Methods. In: Computational Methods for Fluid Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-99693-6_4

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  • DOI: https://doi.org/10.1007/978-3-319-99693-6_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-99691-2

  • Online ISBN: 978-3-319-99693-6

  • eBook Packages: EngineeringEngineering (R0)

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