Numerical methods in fluid mechanics: Difference between revisions

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:<math> \lim_{\Delta x \to 0}f'(x) = \frac {f(x+\Delta x)-f(x)}{\Delta x} </math>

with a finite limiting process, i.e.

:<math> f'(x) =\frac {f(x+\Delta x) - f(x)}{\Delta x} + O(\Delta x) </math>

The term <math>O(\Delta x)</math> gives an indication of the magnitude of the error as a function of the mesh spacing. In this instance, the error is halfedhalved if the grid spacing, _x is halved, and we say that this is a first order method. Most FDM used in practice are at least second order accurate except in very special circumstances. Finite Difference method is still the most popular numerical method for solution of PDEs because of their simplicity, efficiency and low computational cost. Their major drawback is in their geometric inflexibility which complicates their applications to general complex domains. These can be alleviated by the use of either mapping techniques and/or masking to fit the computational mesh to the computational ___domain.

==Finite element method==

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:<math> \frac {u^{n+1} -u^n}{\Delta t } \approx \kappa u^n </math>

Equation is an explicit approximation to the original differential equation since no information about the unknown function at the future time (''n''&nbsp;+&nbsp;1)_''t'' has been used on the right hand side of the equation. In order to derive the error committed in the approximation we rely again on [[Taylor series]].

==Backward difference==

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==References==

'''Sources'''

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

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

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

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

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

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

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

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

'''Citations'''

{{Reflist}}

[[Category:Computational fluid dynamics]]