Neumann Condition on Curved Boundary

[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

 

[Achy47]

for your question! Applying boundary conditions on curved boundaries can be tricky, but it is definitely possible to do so with second order accuracy. Here are some steps that you can follow to achieve this:

1. Use a curvilinear coordinate system: Instead of using a regular Cartesian coordinate system, you can use a curvilinear coordinate system where the boundaries are described by a smooth, continuous curve. This will make it easier to define the boundary conditions and maintain accuracy.

2. Use a higher order interpolation scheme: Instead of using bilinear interpolation, you can use a higher order interpolation scheme such as bicubic or biquadratic interpolation. This will give you a more accurate representation of the curved boundary and help maintain second order accuracy.

3. Use ghost points: Another approach is to use ghost points, where you create additional grid points outside the domain to represent the curved boundary. These points can be used to interpolate the boundary conditions and maintain second order accuracy.

4. Check your discretization scheme: Make sure that your discretization scheme for the Poisson equation is also second order accurate. If your discretization scheme is only first order accurate, then even with accurate boundary conditions, your overall solution will only be first order accurate.

5. Consider using other numerical methods: If you are still having trouble achieving second order accuracy with the Finite Difference Method, you can consider using other numerical methods such as Finite Element Method or Spectral Method. These methods are better suited for handling curved boundaries and can provide higher order accuracy.

I hope these suggestions help you in solving your problem. Here are some additional resources that you may find helpful:

- "Numerical Methods for Partial Differential Equations" by William F. Ames
- "Finite Difference Methods for Ordinary and Partial Differential Equations" by Randall J. LeVeque
- "Numerical Recipes: The Art of Scientific Computing" by William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery

Best of luck with your research!