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 non-uniform 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

Deriving them can be done using taylor series expansion.

Assume you want to find the derivative at node i using the points i-1,i,i+1.

fx_i = Af_i + B f_i-1 + C f_i+1

your aim is to find A,B,C now.Expand f_i+1 and f_i-1 using taylor expansion and assume (hn=abs(x_i-x_i-1) , hp=abs(x_i+1-x_i) )

f_i-1=f_i-hn 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.

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

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