  We provide a primer to numerical methods based on Taylor series expansions
such as generalized finite difference methods and collocation methods. We
provide a detailed benchmarking strategy for these methods as well as all data
files including input files, boundary conditions, point distribution and
solution fields, so as to facilitate future benchmarking of new methods. We
review traditional methods and recent ones which appeared in the last decade.
We aim to help newcomers to the field understand the main characteristics of
these methods and to provide sufficient information to both simplify
implementation and benchmarking of new methods. Some of the examples are chosen
within a subset of problems where collocation is traditionally known to perform
sub-par, namely when the solution sought is non-smooth, i.e. contains
discontinuities, singularities or sharp gradients. For such problems and other
simpler ones with smooth solutions, we study in depth the influence of the
weight function, correction function, and the number of nodes in a given
support. We also propose new stabilization approaches to improve the accuracy
of the numerical methods. In particular, we experiment with the use of a
Voronoi diagram for weight computation, collocation method stabilization
approaches, and support node selection for problems with singular solutions.
With an appropriate selection of the above-mentioned parameters, the resulting
collocation methods are compared to the moving least-squares method (and
variations thereof), the radial basis function finite difference method and the
finite element method. Extensive tests involving two and three dimensional
problems indicate that the methods perform well in terms of efficiency
(accuracy versus computational time), even for non-smooth solutions.
