An efficient Chebyshev spectral collocation method for the solution of Allen-Cahn equation

ZHANG Rongpei, LIU Jia, WANG Yu

(1. College of Mathematics and Systems Science, Shenyang Normal University, Shenyang 110034, China; 2. Department of Foreign Language, Shenyang Normal University, Shenyang 110034, China)

計算數(shù)學(xué)

An efficient Chebyshev spectral collocation method for the solution of Allen-Cahn equation

ZHANG Rongpei1, LIU Jia2, WANG Yu1

(1. College of Mathematics and Systems Science, Shenyang Normal University, Shenyang 110034, China; 2. Department of Foreign Language, Shenyang Normal University, Shenyang 110034, China)

This paper proposes a new highly accurate numerical method for the solution of Allen-Cahn equation. In space discretization, we use the Chebyshev spectral collocation method which has spectral accuracy. After space discretization we develop the nonlinear ordinary differential equations(ODEs) in matrix formulation. The compact implicit integration factor(cIIF) method is later applied for the nonlinear ODEs. This approach has the advantage that the storage and CPU cost are significantly reduced. Numerical results are presented to demonstrate the accuracy, stability, and efficiency of the method.

Chebyshev spectral collocation method; Allen-Cahn equation; Compact implicit integration factor method

In this paper, we consider the numerical solution of the nonlinear Allen-Cahn equation of the following form:

with the following boundary and initial condition:

The Allen-Cahn equation was first introduced by JohnW. Cahn and his graduate student Sam Allen to describe the motion of anti-phase boundaries in metallicalloys[1]. Specifically, it was proposed as a simple model for phase separation of metallic components within a binary alloy at a fixed temperature[2]. The function u, known as the phase field function, assumes values between -1 and 1 where the endpoints represent volumes with pure states. Similarly, values in the region -1

In recent years, many scholars have proposed a variety of numerical methods for equation(1). There are finite difference methods, finite element method, Fourier spectral method[3-6]. Compared with the finite difference method, the low order precision of the finite element method, the spectral method can achieve the convergence of the exponential order with only a small number of nodes and the discretization of the suitable orthogonal polynomials. Commonly used spectral collocation methods are mainly the Fourier collocation method for the periodic boundary value problem[7], the Chebyshev collocation method for general boundary value problem[8], the Hermite collocation method on infinite region and so on[9]. In this paper, the solution region of equation (1) is set as a finite region and the homogeneous Neumann boundary condition (2) is considered. The Chebyshev spectral collocation method is used to solve (1).

After the spatial discretization of the equation (1), we obtain the nonlinear ordinary differential equations (ODEs). The time discretization method can usually be divided into two categories: one is the method of explicit methods such as Runge-Kutta method, linear multi-step method and so on. The method does not need to form the total stiffness matrix. However, since the Allen-Cahn equation group (1) is rigid, it has a strict constraint on the explicit time step. The other method is implicit Runge-Kutta method, backward Euler method and other implicit method. Although implicit method can allow a large time step, it is difficult to solve the large nonlinear algebraic equations. If we use the Newton iterative method, we need to calculate the Jacobian matrix which is a great challenge for the implicit method.

There are also other efficient time discrete formats, such as the Alternating direction implicit method (ADI). We will point out here that the ADI method is not applicable here because the spectral differential matrix obtained by the spectral configuration method is full and can not use the fast solution of algebraic linear equations. In this paper, we choose the compact implicit integration factor method to solve the discrete-formed non-linear ODEs. This method was developed on the basis of the implicit integral factor method proposed by Nie et al in 2006[10]. The traditional implicit integral factor method encounter some difficulties in solving the high dimension problem, mainly because the storage capacity and the computation amount of the exponential operation are very large. It leads to the decrease of the operation speed. The compact implicit factor method greatly reduces the CPU memory and improves the computation speed by introducing the compact expression of the discrete matrix and exponentially the matrix in each direction[11].

The text is arranged as follows: section 1 constructs the Chebyshev spectral matrix of the second derivative and discretizes the nonlinear Allen-Cahn (1) to form a matrix form. And Section 2 gives the compact implicit integral method time discrete scheme to solve the nonlinear ordinary differential equations. Section 3 gives a numerical example to verify the effectiveness of the algorithm. Finally, a summary of this paper is given.

1 Chebyshev spectral collocation method

1.1 Spectral differential matrix

TheTk(x) is defined as the standardkorder Chebyshev polynomial over [-1,1], which satisfies the following recurrence relation:

Given theN+1 Gauss-Lobatto points on [-1,1] as the collocation point:

Set theNthorder polynomialuN(x)∈PNin the above collocation pointxjmeetuN(xj)=u(xj), then there is

Among them,hj(x) is the Lagrange basis function. The spectral collocation method requires that the differential equation (1) satisfies the equation at the collocation point. In order to solve the derivative values at the collocation point, we introduce the Chebyshev spectral differential matrix. Take the p-order derivative on the equation (4), we get

Set the collocation pointxiinto the above formula, then

The second-order spectral differential matrix can also be derived from the interpolation polynomial. For convenience, we obtain the second-order spectral differential matrix directly for the first-order spectral differential matrix,ie,D(2)=(D(1))2.

From the value of the functionu(x) at the collocation point and the spectral differential matrix, the p-order derivative ofu(x) at the collocation point can be written in the form of a matrix vector product as follows:

U(1)=D(1)U,U(2)=D(2)U

1.2 Chebyshev collocation method

whereui,jrepresents the numerical solution ofuat the grid point (xi,xj).

where matrixF(U) is defined asF(U)=f(ui,j).

2 Compact implicit integration factor method

In this section, we solve the nonlinear ordinary differential equations (7) by using the compact implicit integral factor method. Define the time step asτ=Δtand the nth time step istn=nτ,n=0,1,2…. The left of the equation (7) is multiplied by the exponential matrixe-Axt, and the right is multiplied by the the exponential matrixe-Ayt. Equation (7) can be written in the following form:

The above formula, fromtntotn+1within a time step, the time variable integral:

The above formula is approximated by a trapezoidal formula, and we obtain the following second-order compact implicit integral factor:

In the above nonlinear equations (8), the first term at the right end can be obtained by the matrix product. For the second term on the right, we use the following Picard iterative method to solve:

when ‖Un+1,k+1-Un+1,k‖∞<ε, the iteration is terminated, we take the iteration threshold in this paper:ε<10-13.

3 Numerical examples

3.1 Accuracy testing

Example 1 We give an accuracy of the nonlinear equation with exact solution. Consider the following two-dimensional reaction diffusion equation:

whenD=0.2,a=0.1 and the solution region is [0,π]2, its exact solution isu=e-0.1t(cosx+cosy), and the calculate termination time ist=1. The initial condition is determined by the exact solution, and the boundary condition is homogeneous Neumann boundary. First, we test the spatial accuracy of the algorithm, the selection of the time step is small enough(taking Δt=1d-3), and the mesh is divided intoN×Nunits. Table 1 lists the numerical results of the selection of different units, we can find that the error of the format is spectral convergence.

Tab.1 The spatial error of Example 1

3.2 Allen-Cahn equation

In the following, we extend the compact implicit factor spectral scheme (8) to the solution of Allen-Cahn equation. It is well known thatuwhich lies in the spinodal interval is unstable and the growth of instabilities results in phase separation, which is called spinodaldθdecomposition. In this subsection, we perform a spinodal decomposition computation by Chebyshev spectral collocation mehtod for different time steps.

Example 2(Spinodal decomposition): We setε=0.015 and the domainΩ=[0,1]×[0,1] is divided into 128×128 mesh. The initial condition isu(x,y,0)=0.005·rand(x,y) where rand(x,y) is a random number between -1 and 1. Fig. 1 shows snapshots of the solution obtained from the Chebyshev spectral collocation method at different times for Δt=1d-4. The results show that the cIIF method is robust and accurate for a larger time step size, leading to correct morphologies in the phase separation process.

(a) t=10-3; (b) t=3×10-3; (c) t=6×10-3; (d) t=10-2.Fig.1 The solution of Example 2

Example 3(Motion of a circle by its mean curvature): One of the well analyzed solutions of the Allen-Cahn equation is the motion of a circle. Suppose a radially symmetric initial condition is given as follows on the domainΩ=[0,1]×[0,1],

which represents a circle centered at (0.5, 0.5) with a radiusR0=0.25. It is well known that the solution of the Allen-Cahn equation with the initial condition is radially symmetric and the radius of the interfacial circle is shrinking by the rate of the curvature of the circle. We compute this two-dimensional problem withε=0.01 using the second order cIIF methods on a regular 128×128 spatial grid. Fig.2 shows the solutions at different times. It can be found that the circle is shrinking with time evolving.

(a) t=0; (b) t=10-2; (c) t=2×10-2; (d) t=2×10-2.Fig.2 The solution of Example 3

4 Summary

In this paper, the Chebyshev spectral collocation method and the compact implicit integral factor method are used to solve the two-dimensional Allen-Cahn equations with homogeneous Neumann boundary conditions. Therefore, the Chebyshev spectral collocation method proposed in this paper is an efficient and accurate numerical method for large time numerical simulation.

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1673-5862(2017)04-0435-06

Chebyshev譜配置方法求解Allen-Cahn方程

張榮培1, 劉 佳2, 王 語1

(1. 沈陽師范大學(xué) 數(shù)學(xué)與系統(tǒng)科學(xué)學(xué)院, 沈陽 110034; 2. 沈陽師范大學(xué) 大學(xué)外語教學(xué)部, 沈陽 110034)

給出了一種求解Allen-Cahn方程的高精度的數(shù)值計算方法。在空間離散中采用具有譜精度的Chebyshev譜配置方法,得到一組非線性常微分方程組(ODEs)。時間方向上,采用緊致隱式積分因子方法。該方法結(jié)合了譜方法和緊致隱式積分因子方法的特點,具有精度高,穩(wěn)定性好,儲存量小以及計算時間快等優(yōu)點。最后給出的數(shù)值算例驗證了該方法的有效性。

Chebyshev譜配置; Allen-Cahn方程; 緊致隱式積分因子

date: 2017-05-15.

Supported: Project supported by the Foundation of Liaoning Educational Committee(L201604).

Biography: ZHANG Rongpei(1978-),male,was born in Xintai city of Shandong province, associate professor of Shenyang Normal University, doctor.

O241.82DocumentcodeA

10.3969/ j.issn.1673-5862.2017.04.011