A semi-analytical boundary integral method for radial functions with application to Smoothed Particle Hydrodynamics

•Fully general method of evaluating the Shepard factor in two and tree dimensions.•Minimized level of numerical approximation and computational expense.•Derivation and implementation details are provided.•Arbitrary accuracy of the method controlled by a single parameter. An efficient method of numerically integrating radially symmetric functions in two and three dimensions is given. Its main benefits include the lack of need for complex meshing and low computational cost. Both of those are achieved by employing classical integral theorems in order to lower the dimensionality of the problem. We discuss the application of the method in the field of Smoothed Particle Hydrodynamics and provide formulations and an algorithm for easy implementation in existing code. The demonstrated implementation depends only on a single parameter with an intuitive meaning and a mildly more complex error-predicting alternative is briefly discussed. The robustness and accuracy of the method are demonstrated for different problem cases.