Numerical study of three-dimensional Turing patterns using a meshless method based on moving Kriging element free Galerkin (EFG) approach

Abstract

Available online xxxx Keywords: Element free Galerkin (EFG) and movingKriging 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 predator a b s t r a c tIn 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 threedimensional 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.

Diffusive instabilities and spatial patterning from the coupling of reaction–diffusion processes with Stokes flow in complex domains

“…In three dimensions, the number of qualitatively distinct patterns becomes quite large (Shoji et al. 2007; Dehghan & Abbaszadeh 2016), although we are unaware of any such classification of such states applicable to this simple system. Instead, we will emphasize the roles of domain shape and curvature, as well as advection, in the selection and stabilization of different patterns.…”

“…The two-dimensional spot and stripe (labyrinthine) patterns have been shown in a variety of works (Boissonade, Dulos & De Kepper 1995;Shoji, Iwasa & Kondo 2003), and this makes the Schnakenberg kinetics an attractive choice, since the behaviour in the absence of advection is well understood. In three dimensions, the number of qualitatively distinct patterns becomes quite large (Shoji et al 2007;Dehghan & Abbaszadeh 2016), although we are unaware of any such classification of such states applicable to this simple system. Instead, we will emphasize the roles of domain shape and curvature, as well as advection, in the selection and stabilization of different patterns.…”

Simulation of predator–prey system with two-species, two chemicals and an additional chemotactic influence via direct meshless local Petrov–Galerkin method

“…A system of diffusion-reaction equations with application in biology is: where f and F are linear and nonlinear functions, respectively. There are several research works that they numerically investigated model (1.1), for example, finite difference method (Dillon et al , 1994), finite volume method (Shakeri and Dehghan, 2011), finite element method (Barreira et al , 2011; Khaled-Abad and Salehi, 2021; Madzvamuse et al , 2005), element free Galerkin method (Dehghan and Abbaszadeh, 2016; Dehghan et al , 2016), Chebyshev spectral method (Tehseen Saleem and Ali, 2018), Legendre spectral element method (Dehghan and Sabouri, 2013), spectral meshless radial point interpolation approach (Shivanian and Jafarabadi, 2020), local radial basis function (RBF) (Sarra, 2012) and convolutional neural network (Zhu and He, 2022). There are various versions of the predator–prey system, for instance, the following Gause-type model (Gause et al , 1936): where ∏( t ) and u ( t ) denote the population size of prey and predator spaces, respectively.…”

Complex Pattern Formations by Spatial Varying Parameters

“…For time discretization, numerical schemes can be chosen with different convergence behavior. The Runge-Kutta method used in [13,38] and the Crank-Nicolson scheme was employed in [22,23]. The alternating direction implicit Crank-Nicholson (ADI-CN) was applied in [17] to solve two dimensional Riesz space fractional diffusion equations.…”