Remarks on the numerical approximation of Dirac delta functions

We investigate the convergence rate of the solutions of one and two-dimensional Poisson-type PDEs where the Dirac delta function, representing the forcing term, is approximated by several expressions. The goal is to see if the solution to a Poisson's equation converges when solved by a numerical method, with a source or sink approximated using a delta proposed in the literature. We will look at two parameters, how fast it converges, by estimating the order of convergence, and how well, by calculating the error between the analytical form and the numerical result. We investigate smoothed discrete delta functions based on the Immersed boundary (IB) approach, and we revisit their definitions, as in level set methods, by expanding their support for assessing higher-order of convergence in PDE solutions. We developed a Python package utilizing FiPy, a PDE solver based on the finite volume (FV) technique, and accelerated the solver with the AmgX package, a GPU solution. We have observed that when the support is wider, better results may be achieved. Moreover, the overall trends of error and the convergence rate in the 2D configuration differ from the 1D problems.