Numerical solution of a non-classical two-phase Stefan problem via radial basis function (RBF) collocation methods

Dehghan, Mehdi; Najafi, Mahboubeh

doi:10.1016/j.enganabound.2016.07.015

Cited by 26 publications

(2 citation statements)

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“…Smooth ϕ(r, ε) e −(εr) 2 Gaussian GA [2,28,16,31] 1 + (εr) 2 Multiquadric MQ [1,4,6,25] 1 + (εr) 2 β , β ∈ R\N 0 Generalized Multiquadric GMQ [19] 1/ 1 + (εr) 2 Inverse Multiquadric IMQ [6,12,19,34] Piecewise smooth ϕ(r) r β , β / ∈ 2N Radial Potential RP [30] r 2β log(r), β ∈ N Thin Plate Spline TPS [4,32,33] r 2β−1 or r 2β log(r), β ∈ N Polyharmonic Spline PHS [3,11,28,29] Table 1. Some well-known RBFs; ε is the shape parameter.…”

Section: Name Of Rbf Abbreviationmentioning

confidence: 99%

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Stabilizing radial basis functions techniques for a local boundary integral method

Marinelli¹

2023

Revista De La Unión Matemática Argentina

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Radial basis functions (RBFs) have been gaining popularity recently in the development of methods for solving partial differential equations (PDEs) numerically. These functions have become an extremely effective tool for interpolation on scattered node sets in several dimensions. One key issue with infinitely smooth RBFs is the choice of a suitable value for the shape parameter ε, which controls the flatness of the function. It is observed that best accuracy is often achieved when ε tends to zero. However, the system of discrete equations from interpolation matrices becomes ill-conditioned. A few numerical algorithms have been presented that are able to stably compute an interpolant, even in the increasingly flat basis function limit, such as the RBF-QR method and the RBF-GA method. We present these techniques in the context of boundary integral methods to improve the solution of PDEs with RBFs. These stable calculations open up new opportunities for applications and developments of local integral methods based on local RBF approximations. Numerical results for a small shape parameter that stabilizes the error are presented. Accuracy and comparisons are also shown for elliptic PDEs.

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Section: Name Of Rbf Abbreviationmentioning

confidence: 99%

“…Applications of the RBF-QR to Stefan problems were presented in [6]. In [2] this algorithm was used in the method of fundamental solutions for boundary value problems and in [7] for the numerical calculation of surface divergence-free RBFs on the sphere.…”

Section: Name Of Rbf Abbreviationmentioning

confidence: 99%