I would like to solve a PDE arising from a variational problem related to a mass conservation problem plus regularization terms. The preferred solution method is finite differences due to the simplicity of implementation. The application area essentially is the extraction of flow information from image sequences depicting fluid motion in curved tubular structures (vessels).
The question I would like to address is how to handle the domain boundaries within a FD formulation appropriately where the boundary is given by a 2D curve that is not necessarily aligned to the uniform grid.
So far I came across the Immersed Boundary method that as far as I understand introduces additional force terms to the PDE dependent on a set of arbitrarily placed points that describe the boundary.
My question is essentially if the IB method is the proper approach to this problem, or are there other possibilities that work with FD and my be an alternative? Additionally if anyone knew about an example C/MATLAB code for IB, that would certainly be of help.
In case, to make the problem simpler, and a boundary with edge aligned segments is considered is there a recommended way to handle the BCs? For rectangular boundaries the addition of "ghost points" (virtual domain points for which setting fixed values at each iteration depending on the values of neighboring cells at the previous step enable the handling of common BC types in an elegant manner) seems as a good solution, however for domains containing "step" boundary segments the approach cannot be directly applied, as a ghost cell may have multiple neighbours within the domain and thus its value cannot be set to satisfy multiple BCs at once. Is there an established way of using such ideas with non rectangular domains?
Thanks, and looking forward to your feedback, Peter