Numerical study of three-dimensional Turing patterns using a meshless method based on moving Kriging element free Galerkin (EFG) approach. (English) Zbl 1359.65199
Summary: In this paper a numerical procedure is presented for solving a class of three-dimensional Turing system. First, we discrete the spatial direction using element free Galerkin (EFG) method based on the shape functions of moving Kriging interpolation. Then, to achieve a high-order accuracy, we use the fourth-order exponential time differencing Runge-Kutta (ETDRK4) method. Using this discretization for the temporal dimension, we obtain an explicit scheme and do not need to solve nonlinear system of equations. The EFG method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the EFG method test and trial functions are moving least squares approximation (MLS) shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Also building shape functions of MLS approximation is a time consuming procedure. Because of the mentioned reasons we employ the shape functions of moving Kriging interpolation technique which have the mentioned property and less CPU time is required for building them. For testing this method on three-dimensional PDEs, we select some equations and system of PDEs such as Allen-Cahn, Gray-Scott, Ginzburg-Landau, Brusselator models, predator-prey model with additional food supply to predator. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed scheme.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 92C15 | Developmental biology, pattern formation |
Keywords:
element free Galerkin (EFG) and moving Kriging interpolation; fourth-order Runge-Kutta method; mathematical biology and Turing system; Gray-Scott model and Allen-Cahn equation; Ginzburg-Landau model and Brusselator equation; predator-prey model with additional food supply to predatorReferences:
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