Different one sided derivative approximation?
 

In my FVM code, I am trying to approximate the face value of pressure at the boundary wall. Since it is a cell-center code, all pressure values are at the cell's center.

For e.g. at the south/bottom wall,

dp/dy=0.

the simplest approx will be p_s=p(i,j)

higher accuracy from Blazark's CFD book gives p_s=(1/8)*(15*p(i,j)-10*p(i,j+1)+3*p(i,j+2)).

however, it is also possible to use

dp/dy=(1/2h)*(3*p_s-4*p_n+p_nn)

where h=delta, p_n=(p(i,j)+p(i,j+1))/2, p_nn=(p(i,j+1)+p(i,j+2))/2

simplifying, p_s=(2/3)*p(i,j)-(1/3)*p(i,j+1)-(1/6)*p(i,j+2)

based on the no. of values used, they should be of the same accuracy, but why are their values different?

Also, does anyone has similar expression for non-uniform grids?

thanks

Re: Different one sided derivative approximation?
 

They may be of the same order but the error can be different. Remember that the error is of the form Ch^r and different schemes of the same order can have different values for C.

For non-uniforms grids you can fit a polynomial

p(y) = p_s + p₁ y + p₂ y² + ...

and also use

dp/dy(y=0) = 0

to obtain the value of p_s

Re: Different one sided derivative approximation?
 

i think there is a mistake in ur simplification, it should come as Ps = (2/3)*Pi,j + (1/2)*Pi,j+1 - (1/6)*Pi,j+2