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discretization of variable parameters

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Old   June 21, 2011, 17:42
Default discretization of variable parameters
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Dear all,

in a porous layer I have a certain liquid s(x) distribution. The relative permeability is function of it: k(x) = s(x)^3. I need to determine the derivative of k, dk/dx. Should I discretize k:

dk/dx|i = (k(i+1) - k(i-1))/(2h)

or reduce the derivative as follow:

dk/dx|i = 3*s(i)^2 * ds/dx|i = 3*s(i)^2*((s(i+1) - s(i-1))/(2h)) ?

They lead to different results, and can not find a valid reason to choose one of them.

thanks and regards

Mauri
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Old   June 25, 2011, 16:44
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Dear friends:
The difference come from the discretisation error.for example,
if we set s(x)=x;
x|i=0.1;
then
dk/dx|i=(x+h)^3-(x-h)^3/(2*h);

using the reduced form we have

dk/dx|i=3*x^2*ds/dx=3*x^2;

if we set h=0.1, the first one gives a value of 0.40 while the second one gives 0.030.
if we reduce the h to 0.01, then the first one gives a avalue of 0.0301;
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