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For second or first order accurate schemes, to what extent, the results are valid?

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Old   July 7, 2013, 11:34
Default For second or first order accurate schemes, to what extent, the results are valid?
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Albert Tong
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Hi forum,

I understand it is not the right place to ask this, but I trust the forum more than textbooks

I am confused by the numerical schemes in numerical simulations in general. For instance, within icoFoam standard solver, the discretisation schemes are,

ddt Euler-----------------------------------1st order;
grad Gauss linear-------------------------2nd order;
div Gauss linear- -------------------------2nd order;
laplacian Gauss linear corrected---------2nd order;
interpolation linear-----------------------2nd order;

So as a whole, we will have 1st order accuracy. If the tolerance is set to be 1e-06, delta_t to be 0.001s and delta_x to be 0.001m, then in the tutorial example of cavity, to what extent the resulting U and p is valid (assuming the convergent is achieved)?

Is it true that anything after the third digit is generally not accurate (since terms of higher than delta_x^2 and delta_t^1 are omitted in Taylor series expansion)?

Many thanks.
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Albert
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Old   July 7, 2013, 17:47
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Lieven
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Hi Albert,

The discretisation schemes that are used in the solver are fully your choise. So you can indeed select 2nd order schemes for the spatial discretization (grad, div, laplacian ...) by selecting a 'Gauss linear' scheme but you might as well select 'upwind' resulting in a 1st order accuracy. But similarly, you can have a 2nd order accuracy of the time integration by setting
Code:
ddt(...)         CrankNicolson 1.0;
(or CrankNicholson, depending on you OF version).

Quote:
Is it true that anything after the third digit is generally not accurate ...
No, this statement is just incorrect. If this were true, there would not be a need for using very fine meshes because regardless the mesh you take, the solution would not improve after the third digit according to this statement... The same holds for the time integration.

The order basically expresses the rate at which the error reduces:
1st order scheme: dT/2 -> error/2
2nd order scheme: dT/2 -> error/4
so the order does not say anything about the absolute value of the error.

Hope this clarifies things a bit.

Cheers,

L
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