Investigations on several numerical methods for the non-local Allen–Cahn equation

Zhai, Shuying; Weng, Zhifeng; Feng, Xinlong

doi:10.1016/j.ijheatmasstransfer.2015.03.071

Cited by 42 publications

(23 citation statements)

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“…Authors of [13] presented a new linearized high-order compact difference method for numerical simulation of threedimensional (3D) Allen-Cahn equation with three kinds of boundary conditions. Authors of [14] investigated some numerical methods for solving the non-local Allen-Cahn equation with a space-time dependent Lagrange multiplier. To this end, they applied [14] several types of methods, including the Crank-Nicolson finite difference method, the finite difference operator splitting method and the Fourier spectral operator splitting method.…”

Section: )mentioning

confidence: 99%

“…Authors of [14] investigated some numerical methods for solving the non-local Allen-Cahn equation with a space-time dependent Lagrange multiplier. To this end, they applied [14] several types of methods, including the Crank-Nicolson finite difference method, the finite difference operator splitting method and the Fourier spectral operator splitting method.…”

Section: )mentioning

confidence: 99%

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

Dehghan

Abbaszadeh

2016

Computers & Mathematics with Applications

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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.

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confidence: 99%

Section: )mentioning

confidence: 99%

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

Dehghan

Abbaszadeh

2016

Computers & Mathematics with Applications

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“…To study the phase transitions or mean curvature flows with the constraint of constant volumes enclosed by the evolving curves on surfaces, besides the surface Cahn-Hilliard equation model [3,6,7], we can also extend the conservative Allen-Cahn equation [8][9][10] as follow…”

Section: Introductionmentioning

confidence: 99%

“…The stabilized semi-implicit schemes which add artificial dissipative terms for stability are improved forms for standard semi-implicit schemes. After splitting, the computation becomes more simpler and maximum principle preserving [9,10,22,[27][28][29][30]. However, their stabilities are achieved by sacrificing the accuracy [4, 12-14, 17, 21].…”

Section: Introductionmentioning

confidence: 99%

“…But for differential equations on complicated surfaces, the meshes may not be uniform, a flexible spatial discretization such as finite element method [3,34] should be used instead of finite difference method. For the conservative Allen-Cahn equation, based on the proposed two schemes for the standard surface Allen-Cahn equation, an operator splitting framework is applied to treat the mass conservation term [9,10,30]. The lumped mass technique has been studied for surface parabolic equations recently [38,39].…”

Section: Introductionmentioning

confidence: 99%

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Unconditionally maximum principle preserving finite element schemes for the surface Allen–Cahn type equations

Xiao

Feng

2019

Numerical Methods Partial

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In this paper, we present two types of unconditionally maximum principle preserving finite element schemes to the standard and conservative surface Allen-Cahn equations. The surface finite element method is applied to the spatial discretization. For the temporal discretization of the standard Allen-Cahn equation, the stabilized semi-implicit and the convex splitting schemes are modified as lumped mass forms which enable schemes to preserve the discrete maximum principle. Based on the above schemes, an operator splitting approach is utilized to solve the conservative Allen-Cahn equation. The proofs of the unconditionally discrete maximum principle preservations of the proposed schemes are provided both for semi-(in time) and fully discrete cases. Numerical examples including simulations of the phase separations and mean curvature flows on various surfaces are presented to illustrate the validity of the proposed schemes. KEYWORDSconvex splitting scheme, lumped mass finite element method, maximum principle preservation, operator splitting approach, stabilized semi-implicit scheme, surface Allen-Cahn type equation 1 Numer Methods Partial Differential Eq. 2020;36:418-438. wileyonlinelibrary.com/journal/num

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Numerical approximation of the conservative Allen–Cahn equation by operator splitting method

Weng

Zhuang

2017

Math Methods in App Sciences

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In this paper, a second-order fast explicit operator splitting method is proposed to solve the mass-conserving Allen-Cahn equation with a space-time-dependent Lagrange multiplier. The space-time-dependent Lagrange multiplier can preserve the volume of the system and keep small features. Moreover, we analyze the discrete maximum principle and the convergence rate of the fast explicit operator splitting method. The proposed numerical scheme is of spectral accuracy in space and of second-order accuracy in time, which greatly improves the computational efficiency. Numerical experiments are presented to confirm the accuracy, efficiency, mass conservation, and stability of the proposed method.In this section, we present an FEOS method to simulate the asymptotic behavior of solution for Eq. (1). The proposed method is based on Fourier spectral method for the space and second-order Strang splitting scheme for the time. For simplicity, we only consider twodimensional (2D) space. The 1D/3D case is defined analogously. In order to illustrate the main idea of the proposed method, we adopted the following definition.

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Investigations on several numerical methods for the non-local Allen–Cahn equation

Cited by 42 publications

References 24 publications

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

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

Unconditionally maximum principle preserving finite element schemes for the surface Allen–Cahn type equations

Numerical approximation of the conservative Allen–Cahn equation by operator splitting method

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