Taylor-Series Expansion Based Numerical Methods: A Primer, Performance Benchmarking and New Approaches for Problems with Non-smooth Solutions

Abstract

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.

A unified algorithm for the selection of collocation stencils for convex, concave, and singular problems

“…These methods can be applied to problems with smooth and non-smooth solution. 2 Many collocation methods have been proposed over the years. The most famous ones are the smooth particle hydrodynamics method (SPH), 3 the reproduced kernel particle method (RKPM), 4 the moving least squares method (MLS), [5][6][7] and the generalized finite difference (GFD) method.…”

“…It is common to most collocation methods and key to ensure an optimum approximation of the field derivatives. 2 The number of support nodes selected for each collocation node may depend on the location of the collocation node in the domain. It has been shown in Reference 2 that increasing the number of support nodes for collocation nodes on the boundary of the domain helps reducing significantly the observed error while maintaining the fill of the stiffness matrix reasonably low.…”

“…For singular problems, the visibility criterion, introduced as part of the element free Galerkin (EFG) method, 14 has been proven effective for both meshless methods [14][15][16] and collocation methods. 2,[17][18][19] The concept of the visibility criterion is presented in Figure 1(A). Considering a domain Ω, only the support nodes X p which are "visible" from the collocation nodes X c are considered in the stencil.…”

“…For a collocation method based on the GFD method, the use of the diffraction criterion did not lead to a lower error that the visibility criterion. 2 Also, the diffraction and transparency criteria cannot be readily applied to all type of convex problem. Therefore, the visibility criterion has been preferred in this work.…”

“…These methods are used to approximate the solution of partial differential equation problems over defined domains. These methods can be applied to problems with smooth and non‐smooth solution 2 . Many collocation methods have been proposed over the years.…”