Evaluating Stochastic Gradient with Random Grid

[Heimdall]

Hi,

I have a random grid, meaning that each cell consists of a random number. I need to evaluate the gradient.

I've tried to apply a basic Euler formula (u_(i+1) - u_(i-1))/2dx but since the values can fluctuate a lot, fluctuations are even stronger for the gradient...
I'm thinking about using a "smoother" method like a five point stencil, which could be better to avoid strong fluctuations... but then I can't find out how to deal with grid boundaries (for Euler method I use (u_(i+1)-u(i))/dx or (u_(i)-u_(i-1))/dx )

How would you do, is there a specific method for this kind of problem ?

Thanks a lot

 

[makc]

how to deal with grid boundaries

how about $$u_{-1}:=2u_{0}-u_{1}$$, etc? I mean (linear) extrapolation.