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m →Finite difference method: clean up spacing around commas and other punctuation fixes, replaced: ,i → , i |
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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>
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The [[CPU]] time to solve the system of equations differs substantially from method to method. Finite differences are usually the cheapest on a per grid point basis followed by the finite element method and spectral method. However, a per grid point basis comparison is a little like comparing apple and oranges. Spectral methods deliver more accuracy on a per grid point basis than either [[Finite element method|FEM]] or [[Finite difference method|FDM]]. The comparison is more meaningful if the question is recast as ”what is the computational cost to achieve a given error tolerance?”. The problem becomes one of defining the error measure which is a complicated task in general situations.
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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'' + 1)<sub>''t''</sub> 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.
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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]]
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