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The meshless local collocation method for solving multi-dimensional Cahn-Hilliard, Swift-Hohenberg and phase field crystal equations. (English) Zbl 1403.74285
Summary: The collocation technique based on the radial basis functions (RBFs) method is simple and efficient for solving a wide area of problems. But the mentioned technique is poor for solving problems that have shock (advection problems) or the discontinuous initial condition. The local RBFs collocation technique is a meshless method based on the strong form. The use of local collocation RBFs method overcomes the mentioned important issue. In the current paper, based on the proposed idea in [B. Wang, Eng. Anal. Bound. Elem. 50, 395–401 (2015; Zbl 1403.65178)], we consider a linear combination of shape functions of local radial basis functions collocation method and moving Kriging interpolation technique. For showing the efficiency of new technique, some multi-dimensional problems such as Cahn-Hilliard, Swift-Hohenberg and phase field crystal equations have been chosen. Moreover, several test problems are given that show the acceptable accuracy and efficiency of the proposed scheme.

74S25 Spectral and related methods applied to problems in solid mechanics
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
74N05 Crystals in solids
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
82D25 Statistical mechanics of crystals
82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
Full Text: DOI
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