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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.

MSC:
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)
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