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

• Published in: Engineering analysis with boundary elements Vol. 72; pp. 111 - 127
• Main Authors: Dehghan, Mehdi, Najafi, Mahboubeh
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.11.2016
• Subjects: Accuracy; Boundaries; Boundary immobilization method; Chebyshev approximation; Chebyshev spectral collocation method; Collocation methods; Extinction; Kansa method; Legendre spectral collocation method; Mathematical analysis; Mathematical models; Radial basis function; RBF–PS method; RBF–QR method; The method of radial basis functions (RBFs); Two-phase Stefan problem; Chebyshev spectral collocation method; 80A22; Boundary immobilization method; Kansa method; 65M70; 65M06; RBF–PS method; Collocation methods; RBF–QR method; Two-phase Stefan problem; The method of radial basis functions (RBFs); Legendre spectral collocation method
• ISSN: 0955-7997; 1873-197X
• DOI: 10.1016/j.enganabound.2016.07.015

The aim of this paper is to make a comparative study of some high order methods for the numerical solution of a non-classical one-dimensional two-phase Stefan problem. The moving boundary is captured explicitly via boundary immobilization method. The Chebyshev and Legendre spectral collocation methods as high order mesh-based techniques and some radial basis function (RBF) collocation techniques as high order meshless methods are used for spatial discretization. The considered Stefan problem has two stages: one before the extinction time (0≤t≤tm) and one after the extinction time (tm≤t). For this particular model there exists a closed form solution for the former stage but there is no analytical solution for the latter one. Numerical results show that RBF-QR method can attain the accuracy of spectral methods when implemented on Chebyshev grid. The high order accuracy for the two stages shows the superiority of the proposed methods in comparison to the previous works.