Simple immersed boundary matrix-cut-method for cartesian grids using FV and FD discretizations

This work presents a means of implementing a simple implicit Cartesian grid matrix-cut method to model the heat and mass transfer in irregular geometries using any implicit Cartesian grid solver in finite-volume and finite-difference formulations. This method allows the precise imposing of Dirichlet, Neumann, and Robin boundary conditions in any internal node of the grid. The main idea of this method comes from a finite element method where the coefficients of the implicit matrix of a discretized equation are modified directly to ensure the consistency of the presence of the boundary conditions at immersed boundaries represented by grid nodes. One of the advantages of this method is its capability to solve a transport equation implicitly in steady and unsteady modes, applied to complex geometry immersed into the Cartesian grid. This method can be easily implemented in any direct matrix solver, e.g., Gauss elimination, or any iterative matrix solver. To validate this method, different cases are considered.