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Back at the university, I got excited about one of the methods used for computing various quantities from electro magnetics. Particularly the method of finite differences, which is not among the widely used methods, but is somewhat simple to understand and compute (Its not the most efficient method either – but it can be easily accelerated by CUDA). It can be used to calculate the distribution of electric field within a well-defined shape such as in a waveguide.The method uses the Laplace’s Equation deduced from the third Maxwell’s Equation when d/dt = 0 (Static Field) and is a special case of the Poisson’s equation without any free charges in space. Used derivation is depicted below:

The Idea behind this method is quite simple. The area of interest is discretized into a (Usually) rectangular grid of points with a distance between them of “h”. Inside the grid, we define the initial conditions such as voltages on various surfaces (IE Borders of the waveguide or voltages of microstrip lines) These are known to be “Dirichlet Conditions“. However we would like to calculate the distribution of the field (The Electric Potential φ) in the discretized area. There is a method to directly solve the problem via matrix inversion, but due to additional complexity, it is usually solved iteratively with one of the popular methods (Jacobi, Gauss-Seidel, SOR).