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October 12, 2007, 11:47 
Finite difference schemes on non uniform grid

#1 
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Hi
i have a non uniform grid, and i would like to use a finite difference scheme upon it to solve a collection of coupled PDE's [i.e. use FD method to create a load of ODE's then solve these using Runge Kutta  Method of Lines]. But i am struggling to find a source for finite difference schemes with nonuniform grids, for approximating y_x and y_xx. I have found formulas for central differences, but would also like formulas for backward and foward differences, for both y_x and y_xx  does antone know of a reference that contains these? Or perhaps you could show me how to derive them?! Thanks in advance, Charlie Ryan 

October 12, 2007, 14:45 
Re: Finite difference schemes on non uniform grid

#2 
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Deriving them can be done using taylor series expansion.
Assume you want to find the derivative at node i using the points i1,i,i+1. fx_i = Af_i + B f_i1 + C f_i+1 your aim is to find A,B,C now.Expand f_i+1 and f_i1 using taylor expansion and assume (hn=abs(x_ix_i1) , hp=abs(x_i+1x_i) ) f_i1=f_ihn fx_i+hn^2/2 fxx_i + HOT f_i+1= f_i +hp fx_i +hp^2/2 fxx_i + HOT substituting back you would get fx_i=(A+B+C)f_i+(B hn + C hp) fx_i +(B hn^2/2 + C hp^2/2)fxx_i equating both sides u get three equations A+B+C =0 B hn + C hp = 1 B hn^2 + C hp^2 =0 Solve to get A,B,C and you have the expansion. 

October 13, 2007, 11:13 
Re: Finite difference schemes on non uniform grid

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A very good book for finite difference is:
"Computational Fluid Mechanics and Heat Transfer" by Dale A. Anderson, John C. Tannehill, R. H. Pletcher I hope this could help you 

October 18, 2007, 11:32 
Re: Finite difference schemes on non uniform grid

#4 
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Hi
thanks for your responses  very helpful both of them. One Q  how do you calculate the error [the HOT?], from the finite difference approximation? Thanks Charlie 

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