Neumann Condition on Curved Boundary using Finite Difference
I am trying the solve the Poisson equation in a domain with curved boundaries using the Finite Difference Method (second order accurate). I need to apply the Neumann condition on the curved boundary. I have used bilinear interpolation to do this but this causes the resultant scheme to be only first order accurate. Could someone please tell me how one may apply the Neumann condition on a curved boundary with second order accuracy? Any links to material would be greatly appreciated!
you can find a second order FD scheme (left or right) in any good CFD book. Check the book of Ferziger and Peric, especially chapter 7. The idea is to fit a 3rd order polynomial through 3 points of your grid (first point on the boundary and the next 2 inside your mesh).
Thanks. I did try using two different interpolations:
w = ax^2 + bxy + cy^2 + dx + ey + f
and w = ax + bxy + cy + d (bilinear interpolation)
They both work but cause my scheme to converge at 'first order' though my finite difference scheme is of second order.
I will try what you just said and see if it works.
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