A global meshless collocation particular solution method for solving the two-dimensional Navier–Stokes system of equations

The method of approximate particular solutions for the time-fractional diffusion equation with a non-local boundary condition

“…Unlike the traditional two-stage approach, where both particular solutions and homogeneous solution are required to reach the final solution, the MAPS used directly RBFs approximation in order to solve PDEs without the need of finding homogeneous solutions. It should be noted that the MAPS has been recently developed to solve various types of partial differential equations, see, e.g., [36,35,29,37,38] and the references therein. In the following, we describe the method briefly.…”

The control volume radial basis function method CV-RBF with Richardson extrapolation in geochemical problems

“…, are real coefficients to be found by solving a collocation system of linear equations as explained in Bustamante et al(2013); Bustamante et al (2014), Orsini et al (2008) and Kansa (2000), Ψ is an RBF and NP is the number of polynomial terms P M 1 − . Although Micchelli (1986) proved that the collocation matrix resulting from the RBF interpolation is always non-singular, its condition number might be large (Schaback, 1995) for the interpolation over the whole domain, and so a stencil should be used to circumvent this problem (Orsini et al, 2009;Bustamante et al, 2014).…”

“…However, the application of RBF transcends the interpolation because they can also be used as the basis of several meshless collocation approaches for solving partial differential equations (PDEs) (see Kansa, 2000, andJumarhon et al, 2000). The most tested approaches so far are global methods that approximate the differential operator over the whole domain (Bustamante et al , 2013), local methods that approximate the PDE locally on a stencil made of some nodes (Bustamante et al, 2014), unsymmetrical methods that directly approximate the solution variable (Kansa, 2000), symmetric methods that achieve a Hermite interpolation with the RBF and the differential operator (Stevens et al, 2009) and finally some methods that approximate the particular solution by RBF (Chen et al, 2012;Bustamante et al, 2014). Although the global formulation becomes unpractical when the number of collocation points is relative large, the local implementation can be used for the improvement of classical numerical methods such as the control volume method CV (Versteeg and Malalasekera, 2007).…”

Numerical solutions of two-dimensional flow fields by using the localized method of approximate particular solutions

“…Recently, the MAPS has been adopted for numerical solutions of wave equation [14], inhomogeneous three-dimensional elliptic equations [15], inverse problems [16], problems of linear elasticity [17], two-dimensional Navier-Stokes equations [18] and anisotropic elliptic problems [19]. Though the MAPS can solve many engineering problems governed by inhomogeneous partial differential equations, the problems of dense and illconditioned matrices limit the MAPS for large-scale problems.…”

“…Thus, in the past decades, many so-called meshless or meshfree methods have been developed for efficiently analyzing various engineering problems, such as the method of fundamental solutions (MFS) [7,8], the Trefftz method (TM) [9,10], the singular boundary method (SBM) [11], the method of approximate particular solutions (MAPS) [12][13][14][15][16][17][18][19], the local radial basis function collocation method (LRBFCM) [20][21][22], the localized method of approximate particular solutions (LMAPS) [23][24][25], the generalized finite difference method (GFDM) [26,27], etc. The TM and the SBM are boundary-type meshless methods, since only boundary nodes are needed in the numerical implementations of these methods.…”