Neumann Condition on Curved Boundary using Finite Difference

[sashankhrao]

Hi,

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!

Thanks

 

[jvicens]

for your post! I have some experience with solving the Poisson equation with curved boundaries using the Finite Difference Method, so I hope I can provide some helpful insights for your problem.

Firstly, it is important to note that applying the Neumann condition on curved boundaries can be more challenging compared to straight boundaries. This is because the normal derivative of the solution at a curved boundary is not constant, unlike at a straight boundary.

To maintain second order accuracy, you will need to use a higher order interpolation scheme instead of bilinear interpolation. This can be done by using higher order polynomials, such as quadratic or cubic, to interpolate the boundary values. These higher order interpolation schemes will give you more accurate values for the normal derivative at the boundary, thus maintaining second order accuracy in your overall solution.

Another approach is to use a staggered grid, where the boundary points are shifted by half a grid spacing. This can help improve the accuracy of the boundary values and the normal derivative. However, this approach may require some modifications to your overall finite difference scheme.

I would also recommend checking out some literature on this topic, such as "Numerical Methods for Partial Differential Equations" by J.W. Thomas or "Finite Difference Methods for Ordinary and Partial Differential Equations" by Randall J. LeVeque. These texts provide a detailed explanation on how to handle Neumann boundary conditions on curved boundaries with second order accuracy.

I hope this helps and good luck with your research!