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I have read some articles about the finite difference method on a cartesian orthogonal grid. I understand how it works when Dirichlet boundary conditions are used, or when Neumann boundary conditions are used on a simple boundary (for example the rectangular boundary of the simulation). What I don't understand is how to use the Neumann boundary conditions in boundaries with arbitrary shape (in a 2-dimensional cartesian orthogonal grid). Suppose I want to simulate the potential flow about a cylinder (in 2D it becomes a circle): can I use the finite difference method on a circular boundary, where that circle has been drawn on the cartesian grid (so it is approximated by a set of little squares)?
Thank you in advance.

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I am not too clear on what is being asked here. Are you asking if you can generate a boundary fitted orthogonal grid? I believe that the curvilinear grid approach would work well for such a case. The idea is to transform the "irregular" physical domain into a computational space which is regular such as a square (in 2D or cube in 3D). After such transformation applying the boundary conditions and discretization of governing equations is trivial. More here –  Isopycnal Oscillation May 3 '13 at 18:35
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Its a method called co-ordinate grid transformations that is used to transform an arbitrarily shaped geometry into a square; in the computational domain. Grid transformations work as parametric transformations do, in co-ordinate geometry. what these transformations basically do is, they map the complex geometry (viz. a curve) into a simpler geometry (a line; in the computational domain) for the computer to handle. Here's a link which mite be useful. A good book on this is CFD by J. Anderson. There's a complete chapter on grid-transformations in this book. A more involved discussion on the same topic.

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