Numerical methods in fluid mechanics: Difference between revisions

Revision as of 14:26, 5 December 2013

Revision as of 22:08, 5 December 2013

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==Discretization==

The central process in CFD is the process of [[discretization]], i.e. the process of taking differential equations with an infinite number of [[degrees of freedom]], and reducing it to a system of finite degrees of freedom. Hence, instead of determining the solution everywhere and for all times, we will be satisfied with its calculation at a finite number of locations and at specified time intervals. The [[partial differential equations]] are then reduced to a system of algebraic equations that can be solved on a computer. Errors creep in during the discretization process. The nature and characteristics of the errors must be controlled in order to ensure that:

<ul><li>* we are solving the correct equations (consistency property)</li>

<li>* that the error can be decreased as we increase the number of degrees of freedom (stability and convergence). </li></ul>

Once these two criteria are established, the power of computing machines can be leveraged to solve the problem in a numerically reliable fashion. Various discretization schemes have been developed to cope with a variety of issues. The most notable for our purposes are: [[finite difference methods]], finite volume methods, [[finite element methods]], and [[spectral methods]].

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==References==

<ol>

<li># Zalesak, S. T., 2005. The design of flux-corrected transport algorithms for structured grids. In: Kuzmin, D., L¨ohner, R., Turek, S. (Eds.), Flux-Corrected Transport. Springer

<li># Zalesak, S. T., 1979. Fully multidimensional flux-corrected transport algorithms for fluids. Journal of Computational Physics.</li>

grids. In: Kuzmin, D., L¨ohner, R., Turek, S. (Eds.), Flux-Corrected Transport.

<li># Leonard, B. P., MacVean, M. K., Lock, A. P., 1995. The flux integral method fofor [[Convection–diffusion equation|multi-dimensional convection]] and diffusion. Applied Mathematical Modelling.</li>

Springer</li>

<li># Shchepetkin, A. F., McWilliams, J. C., 1998. Quasi-monotone advection schemes based on explicit locally adaptive [[dissipation]]. Montlhy Weather Review

<li>Zalesak, S. T., 1979. Fully multidimensional flux-corrected transport algorithms for fluids.

# Jiang, C.-S., Shu, C.-W., 1996. Efficient implementation of weighed eno schemes. Journal of Computational Physics.</li>

<li># Finlayson, B. A., 1972. The Method of Weighed Residuals and Variational Principles. Academic Press.

<li>Leonard, B. P., MacVean, M. K., Lock, A. P., 1995. The flux integral method fo [[Convection–diffusion equation|multi-dimensional convection]] and diffusion. Applied Mathematical Modelling.</li>

<li># Durran, D. R., 1999. Numerical Methods for [[Wave function|Wave Equations]] in Geophysical Fluid Dynamics. Springer, New York.

<li>Shchepetkin, A. F., McWilliams, J. C., 1998. Quasi-monotone advection schemes based

<li># Dukowicz, J. K., 1995. Mesh effects for rossby waves. Journal of Computational Physics</li>

on explicit locally adaptive [[dissipation]]. Montlhy Weather Review</li>

<li># Canuto, C., Hussaini, M. Y., Quarteroni, A., Zang, T. A., 1988. Spectral Methods in Fluid Dynamics. Springer Series in Computational Physics. Springer-Verlag, New York.

<li>Jiang, C.-S., Shu, C.-W., 1996. Efficient implementation of weighed eno schemes. Journal

<li># Butcher, J. C., 1987. The Numerical Analysis of [[Ordinary Differential Equations]]. John Wiley and Sons Inc., NY.

of Computational Physics</li>

<li># Boris, J. P., Book, D. L., 1973. Flux corrected transport, i: Shasta, a fluid transport algorithm that works. Journal of Computational Physics

<li>Finlayson, B. A., 1972. The Method of Weighed Residuals and Variational Principles.

Academic Press.</li>

<li>Durran, D. R., 1999. Numerical Methods for [[Wave function|Wave Equations]] in Geophysical Fluid Dynamics.

Springer, New York.</li>

<li>Dukowicz, J. K., 1995. Mesh effects for rossby waves. Journal of Computational Physics</li>

<li>Canuto, C., Hussaini, M. Y., Quarteroni, A., Zang, T. A., 1988. Spectral Methods in

Fluid Dynamics. Springer Series in Computational Physics. Springer-Verlag, New York.</li>

<li>Butcher, J. C., 1987. The Numerical Analysis of [[Ordinary Differential Equations]]. John

Wiley and Sons Inc., NY.</li>

<li>Boris, J. P., Book, D. L., 1973. Flux corrected transport, i: Shasta, a fluid transport

algorithm that works. Journal of Computational Physics</li>

</ol>

[[Category:Computational fluid dynamics]]