discretization of variable parameters
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
The difference come from the discretisation error.for example,
if we set s(x)=x;
using the reduced form we have
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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