# Efficient Solvers of Discontinuous Galerkin Discretization for the Cahn–Hilliard Equations

## Abstract

In this paper, we develop and analyze a fast solver for the system of algebraic equations arising from the local discontinuous Galerkin (LDG) discretization and implicit time marching methods to the Cahn–Hilliard (CH) equations with constant and degenerate mobility. Explicit time marching methods for the CH equation will require severe time step restriction \((\varDelta t \sim O(\varDelta x^4))\), so implicit methods are used to remove time step restriction. Implicit methods will result in large system of algebraic equations and a fast solver is essential. The multigrid (MG) method is used to solve the algebraic equations efficiently. The Local Mode Analysis method is used to analyze the convergence behavior of the linear MG method. The discrete energy stability for the CH equations with a special homogeneous free energy density \(\Psi (u)=\frac{1}{4}(1-u^2)^2\) is proved based on the convex splitting method. We show that the number of iterations is independent of the problem size. Numerical results for one-dimensional, two-dimensional and three-dimensional cases are given to illustrate the efficiency of the methods. We numerically show the optimal complexity of the MG solver for \(\mathcal{P }^1\) element. For \(\mathcal{P }^2\) approximation, the optimal or sub-optimal complexity of the MG solver are numerically shown.

### Keywords

Cahn–Hilliard equation Local discontinuous Galerkin method Convex splitting method Multigrid algorithm FAS multigrid Additive Runge–Kutta Diagonally implicit Runge–Kutta Local mode analysis### Mathematics Subject Classification (2010)

65M60 35K55## 1 Introduction

Our main interest is to explore an efficient high order implicit time discretization method for the CH equation coupled with the LDG spatial discretization. In this paper, we will apply the implicit additive Runge–Kutta (ARK) method to the CH equation and the numerical results show that it is an efficient time discretization method for the CH equation coupled with the LDG method. The third order ARK method requires to solve three linear systems of equations at each time step. Traditionally iterative methods such as Gauss–Seidel method suffer from slow convergence rates. We will apply the MG method to accelerate the convergence rates when solving the system of equations, which derived by the LDG spatial discretization and high order time marching method for the CH equation. In order to predict the MG behavior, a two-level Local Mode Analysis is used to study the convergence of the MG method.

The mobility \(b(u)\) in the CH equation can be constant or degenerate. For the degenerate mobility, the ARK method with the linear MG solver is not efficient. We will apply the diagonally implicit Runge–Kutta (DIRK) [1] time discretization method to treat the nonlinear CH equation. Then it requires to solve nonlinear systems of algebraic equations at each time step. The nonlinear Gauss–Seidel method and the Newton method can be used to solve the nonlinear equations, but they are not effective for large system. We will apply the nonlinear Full Approximation Scheme (FAS) MG method coupled with the LDG spatial discretization for the CH equation and the numerical results show that the convergence rates of the method is \(O(N)\). We also prove the unconditional energy stability for the first order scheme in time based on the convex splitting method. We numerically show the optimal complexity of the MG solver for \(\mathcal{P }^1\) element. For \(\mathcal{P }^2\) approximation, the optimal or sub-optimal complexity of the MG solver are numerically shown.

The discontinuous Galerkin (DG) method is a class of finite element methods using completely discontinuous piecewise polynomials as the solution and the test spaces. Reed and Hill [26] first introduced the DG method in 1973, in the framework of neutron linear transport. For partial differential equations (PDEs) containing higher than first order spatial derivatives, the DG method can not be applied directly, so the LDG method was introduced. The first LDG method was introduced by Cockburn and Shu [12] for time-dependent convection-diffusion systems. The idea of the LDG method is to rewrite the equations with higher order derivatives as a first order system, then apply the DG method to the system. In [35], the ARK method was explored to solve the stiff ordinary differential equations (ODEs) resulting from a LDG spatial discretization to PDEs with higher order spatial derivatives. For a detailed description about the LDG methods for high-order time-dependent PDEs, we refer the readers to [36]. Recently, the MG method coupled with the DG spatial discretization for the compressible Naiver–Stokes equation [25, 27] and the Euler equation [5, 6] have been studied. In [30, 31], the MG method was introduced to solve the system of algebraic equations arising from the higher order DG discretization of advection dominated flows.

Many numerical methods have been developed to treat the CH equation, using finite elements [2, 3, 4, 7, 8, 13, 14, 15, 17], discontinuous Galerkin methods [11, 18, 32], multigrid method [22, 23, 24] and finite difference methods [16, 19, 28]. Xia et al. [34] developed the LDG methods for the CH equation, which was high order accurate, nonlinear stable and flexible for arbitrary \(h\) and \(p\) adaptivity. The explicit time discretization method was used in this paper and it lead to strict restrict on time step, so implicit methods should be used to improve computational efficiency.

The rest of the paper is organized as following. In Sect. 2, we describe a full-discrete LDG scheme for CH equation and prove the unconditional energy stability based on the convex splitting method. In Sect. 3, we introduce two different high order time discretization methods, the ARK method and the DIRK method. In Sect. 4, we study the convergence of the linear bi-grid algorithm numerically. Section 5 contains numerical results for the nonlinear problems which include the CH equation for one-dimensional, two-dimensional and three-dimensional cases. We give some concluding remarks in Sect. 6. In the “Appendices 1 and 2”, we give a detailed description of the linear MG method and the nonlinear FAS MG method. A fairly complete description of the Local Mode Analysis for the bi-grid algorithm is given in “Appendix 3”.

## 2 The Convex Splitting LDG Method

### 2.1 Notations

Let \(\mathcal{T }_h\) denote a tessellation of \(\varOmega \) with shape-regular element \(K\). Let \(\Gamma \) denote the union of the boundary faces of elements \(K\in \mathcal{T }_h\), i.e. \(\Gamma =\bigcup _{K \in \mathcal{T }_h} \partial K\), and \(\Gamma _0=\Gamma \backslash \partial \varOmega \).

In order to describe the flux functions, we need to introduce some notations. Let \(e\) be a face shared by the “left” and “right” elements \(K_L\) and \(K_R\) (we refer to [36] for more details of the definition). Define the normal vectors \({\varvec{\nu }}_L\) and \({\varvec{\nu }}_R\) on \(e\) pointing exterior to \(K_L\) and \(K_R\), respectively. If \(\psi \) is a function on \(K_L\) and \(K_R\), but possibly discontinuous across \(e\), let \(\psi _L\) denote \((\psi |_{K_L})|_e\) and \(\psi _R\) denote \((\psi |_{K_R})|_e\), the left and right trace, respectively.

### 2.2 Full-Discrete LDG Scheme

### 2.3 Energy Stability

**Proposition 1**

*Proof*

## 3 The High Order Time Discretization Methods

The scheme (2.4) is stable regardless of the time step size \(\varDelta t\). Thus, we say the scheme is unconditionally energy stable, but it is first order accuracy in time. Furthermore, the convex splitting scheme is just for a special choice of \(\varPsi (u) (\varPsi (u)=\frac{1}{4}(1-u^2)^2)\). Our purpose is to obtain a high order scheme in time and for arbitrary \(\varPsi (u)\), so we will explore high order time discretization methods for the CH equation in this section.

### 3.1 The Linearization Scheme

Xia et al. [35] explored the ARK method and found that it was an efficient method when it was coupled with the LDG spatial discretization for solving PDEs containing higher order spatial derivatives. In this section, we apply the ARK method to the CH equation (1.1), which treats the linear part implicitly and the nonlinear part explicitly. But the non-negative mobility \(b(u)\) in the CH equation can be constant or degenerate. For the degenerate mobility \(b(u)\), we can not apply the ARK method directly because of the nonlinear term \(b(u)\), so we consider a linearization technique.

- 1.Rewrite the CH equation (1.1) in the following form$$\begin{aligned} u_t=\nabla \cdot (b(u^n)\nabla (-\gamma \varDelta u))+\nabla \cdot ((b(u)-b(u^n))\nabla (-\gamma \varDelta u))+\nabla \cdot (b(u)\nabla (\varPsi ^{\prime }(u))).\nonumber \\ \end{aligned}$$(3.1)
- 2.Apply the LDG spatial discretization similar to [34] to the equation (3.1) and obtain an ODEwhere \(F_S(t,u(t))\) is obtained by the LDG discretization to \(\nabla \cdot (b(u^n)\nabla (-\gamma \varDelta u))\) and \(F_N(t,u(t))\) is obtained by the remaining terms in (3.1).$$\begin{aligned} {\left\{ \begin{array}{ll} u_t=F_N(t,u(t))+F_S(t,u(t)), &{}t \in [0,T],\\ u(0)=u^0, &{} \end{array}\right. } \end{aligned}$$(3.2)
- 3.
Apply the third order ARK time discretization method to ODE (3.2) and it requires to solve three algebraic non-symmetric linear systems at each time step. What we should have in mind is that the time step \(\varDelta t\) can not be chosen too large because of the explicit treatment of the term \(\nabla \cdot ((b(u)-b(u^n))\nabla (-\gamma \varDelta u))\).

- 4.
Apply the linear MG method to solve the linear system.

### 3.2 The Nonlinear Scheme

After the implicit discretization, it requires to solve three nonlinear systems of algebraic equations (3.4) at each time step. We apply the nonlinear FAS MG method to solve these nonlinear systems, which will be described in detail in “Appendix 2”.

## 4 Local Mode Analysis of the Two-Level Algorithm

### 4.1 Notations

A fairly complete description of the linear MG method and the nonlinear FAS MG method are given in “Appendices 1 and 2”, respectively.

### 4.2 The Convergence of the Two-Grid Algorithm

- One space dimension
- 1)
We choose \(\alpha =0.75\) with Jacobi smoother and \(\alpha =1.0\) (no damping) with Gauss–Seidel smoother for \(\mathcal{P }^1\) and \(\mathcal{P }^2\) approximation from Fig. 1.

- 2)
From Figs. 2 and 3, we can see that the Gauss–Seidel smoother has better convergence behavior than the Jacobi smoother.

- 3)
The MG method is not convergent for \(\mathcal{P }^2\) approximation with damped Jacobi smoother according to Fig. 2.

- 1)
- Two space dimension
- 1)
We choose \(\alpha =0.85\) with Jacobi smoother and \(\alpha =1.0\) (no damping) with Gauss–Seidel smoother for \(\mathcal{P }^1\) and \(\mathcal{P }^2\) approximation from Fig. 4.

- 2)
From Figs. 5 and 6, we can see that the Gauss–Seidel smoother has better convergence behavior than the Jacobi smoother.

- 3)
The MG method is not convergent for \(\mathcal{P }^2\) approximation with damped Jacobi smoother according to Fig. 5.

- 1)

## 5 Numerical Results

In this section, we perform numerical experiments of the LDG method applied to the CH equation. We use the implicit third order ARK time-marching method and the resulting linear system is solved by the linear MG method. The third order DIRK time discretization method is also applied to the CH equation and we solve the resulting nonlinear equations by the nonlinear FAS MG method. A comparison is made among these two different methods. For the spatial discretization we use uniform meshes. In our numerical experiments, the number of pre- and post-relaxations is \(\nu _1=\nu _2=3\).

### 5.1 One Space Dimension

*Example 1*

Convex splitting scheme for degenerate mobility

N | \(\varDelta t=0.1\varDelta x\) | \(\varDelta t=1.0\varDelta x\) | ||||||
---|---|---|---|---|---|---|---|---|

\(L^2 \) error | Order | \( L^\infty \) error | Order | \(L^2 \) error | Order | \( L^\infty \) error | Order | |

16 | 1.19E–01 | – | 1.05E–01 | – | 3.45E–01 | – | 2.07E–01 | – |

32 | 5.43E–02 | 1.14 | 5.03E–02 | 1.07 | 2.02E–01 | 0.77 | 1.18E–01 | 0.80 |

64 | 2.59E–02 | 1.06 | 2.46E–02 | 1.02 | 1.09E–01 | 0.89 | 6.33E–02 | 0.90 |

128 | 1.27E–02 | 1.03 | 1.22E–02 | 1.01 | 5.68E–02 | 0.94 | 3.28E–02 | 0.95 |

\(\varDelta t=5.0\varDelta x\) | \(\varDelta t=10.0\varDelta x\) | |||||||
---|---|---|---|---|---|---|---|---|

\(L^2 \) error | order | \( L^\infty \) error | order | \(L^2 \) error | order | \( L^\infty \) error | order | |

16 | 5.78E–01 | – | 3.22E–01 | – | 5.78E–01 | – | 3.22E–01 | – |

32 | 4.98E–01 | 0.21 | 2.78E–01 | 0.21 | 5.17E–01 | 0.16 | 2.89E–01 | 0.15 |

64 | 3.55E–01 | 0.49 | 1.96E–01 | 0.50 | 4.84E–01 | 0.10 | 2.64E–01 | 0.13 |

128 | 2.24E–01 | 0.67 | 1.23E–01 | 0.67 | 3.52E–01 | 0.47 | 1.94E–01 | 0.44 |

*Example 2*

ARK method for constant mobility

N | \(L^2 \) error | Order | \( L^\infty \) error | Order | |
---|---|---|---|---|---|

\(\mathcal{P }^0\) | 16 | 5.86E–01 | – | 5.11E–01 | – |

32 | 2.75E–01 | 1.08 | 2.47E–01 | 1.04 | |

64 | 1.35E–01 | 1.02 | 1.22E–01 | 1.01 | |

128 | 6.75E–02 | 1.00 | 6.13E–01 | 1.00 | |

\(\mathcal{P }^1\) | 16 | 8.81E–02 | – | 1.09E–01 | – |

32 | 2.17E–02 | 2.02 | 2.71E–02 | 2.01 | |

64 | 5.40E–03 | 2.00 | 6.77E–03 | 2.00 | |

128 | 1.35E–03 | 2.00 | 1.69E–03 | 2.00 | |

\(\mathcal{P }^2\) | 16 | 4.41E–03 | – | 5.93E–03 | – |

32 | 5.40E–04 | 3.03 | 7.66E–04 | 2.95 | |

64 | 6.72E–05 | 3.01 | 9.65E–05 | 2.99 | |

128 | 8.38E–06 | 3.00 | 1.20E–05 | 3.00 |

The number of MG iterations required to reduce the norm of the residual below the tolerance \(\epsilon =1.0 \times 10^{-9}\) for \(\mathcal{P }^1\) approximation in one space dimension

\(\varDelta x\) | Jacobi smoother | Gauss–Seidel smoother |
---|---|---|

\(4\pi /32\) | 8 | 7 |

\(4\pi /64\) | 9 | 8 |

\(4\pi /128\) | 9 | 7 |

\(4\pi /256\) | 9 | 7 |

The number of MG iterations required to reduce the norm of the residual below the tolerance \(\epsilon =1.0 \times 10^{-9}\) for \(\mathcal{P }^2\) approximation with Gauss–Seidel smoother in one space dimension

\(\varDelta x\) | \(4\pi /16\) | \(4\pi /32\) | \(4\pi /64\) |
---|---|---|---|

number of MG iterations | 9 | 9 | 16 |

*Example 3*

ARK and DIRK methods for constant mobility

N | Linear MG | Nonlinear FAS MG | |||||||
---|---|---|---|---|---|---|---|---|---|

\(L^2 \) error | Order | \( L^\infty \) error | Order | \(L^2 \) error | Order | \( L^\infty \) error | Order | ||

\(\mathcal{P }^0 \) | 16 | 1.01E–00 | – | 9.61E–01 | – | 1.01E–00 | – | 9.61E–01 | – |

32 | 5.08E–01 | 0.99 | 4.88E–01 | 0.97 | 5.08E–01 | 0.99 | 4.88E–01 | 0.97 | |

64 | 2.54E–01 | 1.00 | 2.44E–01 | 1.00 | 2.54E–01 | 1.00 | 2.44E–01 | 1.00 | |

128 | 1.27E–01 | 1.00 | 1.22E–01 | 1.00 | 1.27E–01 | 1.00 | 1.22E–01 | 1.00 | |

\(\mathcal{P }^1\) | 16 | 1.24E–01 | – | 1.51E–01 | – | 1.25E–01 | – | 1.51E–01 | – |

32 | 3.11E–02 | 1.99 | 3.75E–02 | 2.00 | 3.13E–02 | 1.99 | 3.74E–02 | 2.01 | |

64 | 7.66E–03 | 2.02 | 9.67E–03 | 1.95 | 7.83E–03 | 2.00 | 9.47E–03 | 1.98 | |

128 | 1.92E–03 | 1.99 | 2.41E–03 | 2.00 | 1.95E–03 | 2.00 | 2.36E–03 | 2.00 | |

\(\mathcal{P }^2 \) | 16 | 2.58E–02 | – | 3.36E–02 | – | 2.56E–02 | – | 3.61E–02 | – |

32 | 3.23E–03 | 2.99 | 4.27E–03 | 2.97 | 3.23E–03 | 2.98 | 4.48E–03 | 3.01 | |

64 | 4.29E–04 | 2.91 | 5.38E–04 | 2.98 | 4.05E–04 | 2.99 | 5.57E–04 | 3.00 | |

128 | 5.31E–05 | 3.01 | 6.47E–05 | 3.05 | 5.08E–05 | 2.99 | 6.97E–05 | 3.00 |

*Example 4*

ARK and DIRK methods for degenerate mobility

N | Linear MG | Nonlinear FAS MG | |||||||
---|---|---|---|---|---|---|---|---|---|

\(L^2 \) error | Order | \( L^\infty \) error | Order | \(L^2 \) error | Order | \( L^\infty \) error | Order | ||

\(\mathcal{P }^0 \) | 16 | 2.69E–01 | – | 2.52E–01 | – | 2.69E–01 | – | 2.52E–01 | – |

32 | 1.34E–01 | 1.00 | 1.26E–01 | 1.00 | 1.34E–01 | 1.00 | 1.26E–01 | 1.00 | |

64 | 6.71E–02 | 1.00 | 6.31E–02 | 1.00 | 6.71E–02 | 1.00 | 6.31E–02 | 1.00 | |

128 | 3.35E–02 | 1.00 | 3.15E–02 | 1.00 | 3.35E–02 | 1.00 | 3.15E–02 | 1.00 | |

\(\mathcal{P }^1 \) | 16 | 1.99E–02 | – | 2.77E–02 | – | 1.99E–02 | – | 2.77E–02 | – |

32 | 4.92E–03 | 2.01 | 6.86E–03 | 2.01 | 4.93E–03 | 2.01 | 6.86E–03 | 2.01 | |

64 | 1.22E–03 | 2.01 | 1.70E–03 | 2.01 | 1.22E–03 | 2.01 | 1.70E–03 | 2.01 | |

128 | 3.06E–04 | 1.99 | 4.26E–04 | 1.99 | 3.06E–04 | 1.99 | 4.26E–04 | 2.00 | |

\(\mathcal{P }^2\) | 16 | 2.21E–03 | – | 3.49E–03 | – | 3.29E–03 | – | 5.21E–03 | – |

32 | 2.77E–04 | 2.99 | 4.39E–04 | 2.99 | 4.13E–04 | 2.99 | 6.55E–04 | 2.99 | |

64 | 3.47E–05 | 2.99 | 5.49E–05 | 3.00 | 5.17E–05 | 3.00 | 8.20E–05 | 3.00 | |

128 | 4.36E–06 | 2.99 | 6.92E–06 | 2.99 | 6.47E–06 | 3.00 | 1.02E–05 | 3.00 |

### 5.2 Two Space Dimension

*Example 5*

Convex splitting scheme for degenerate mobility

N | \(\varDelta t=0.1\varDelta x\) | \(\varDelta t=1.0\varDelta x\) | ||||||
---|---|---|---|---|---|---|---|---|

\(L^2 \) error | Order | \( L^\infty \) error | Order | \(L^2 \) error | Order | \( L^\infty \) error | Order | |

16 | 2.23E–02 | – | 4.40E–02 | – | 1.29E–01 | – | 2.50E–01 | – |

32 | 9.81E–03 | 1.18 | 1.95E–02 | 1.17 | 7.72E–02 | 0.75 | 1.52E–01 | 0.71 |

64 | 4.56E–03 | 1.10 | 9.10E–03 | 1.10 | 4.05E–02 | 0.93 | 8.08E–02 | 0.92 |

128 | 2.20E–03 | 1.05 | 4.39E–03 | 1.04 | 2.06E–02 | 0.97 | 4.13E–02 | 0.97 |

\(\varDelta t=5.0\varDelta x\) | \(\varDelta t=10.0\varDelta x\) | |||||||
---|---|---|---|---|---|---|---|---|

\(L^2 \) error | Order | \( L^\infty \) error | Order | \(L^2 \) error | Order | \( L^\infty \) error | Order | |

16 | 2.53E–01 | – | 4.75E–01 | – | 2.53E–01 | – | 4.75E–01 | – |

32 | 2.36E–01 | 0.10 | 4.59E–01 | 0.05 | 2.47E–01 | 0.04 | 4.65E–01 | 0.03 |

64 | 1.55E–01 | 0.61 | 3.07E–01 | 0.58 | 2.34E–01 | 0.08 | 4.58E–01 | 0.02 |

128 | 9.14E–02 | 0.76 | 1.82E–01 | 0.75 | 1.54E–01 | 0.60 | 3.07E–01 | 0.58 |

The number of FAS MG iterations required to reduce the norm of the residual below the tolerance \(\epsilon =1.0 \times 10^{-9}\) for \(\mathcal{P }^1\) approximation in two space dimension

\(\varDelta x\) | Jacobi smoother | Gauss–Seidel smoother |
---|---|---|

\(2\pi /32\) | 13 | 8 |

\(2\pi /64\) | 12 | 8 |

\(2\pi /128\) | 10 | 8 |

\(2\pi /256\) | 11 | 8 |

The number of FAS MG iterations required to reduce the norm of the residual below the tolerance \(\epsilon =1.0 \times 10^{-9}\) for \(\mathcal{P }^2\) approximation with Gauss–Seidel smoother in two space dimension

\(\varDelta x\) | \(2\pi /16\) | \(2\pi /32\) | \(2\pi /64\) |
---|---|---|---|

number of MG iterations | 9 | 11 | 16 |

*Example 6*

ARK and DIRK methods for constant mobility

*Example 7*

ARK and DIRK methods for degenerate mobility

### 5.3 Three Space Dimension

*Example 8*

Convex splitting scheme with degenerate mobility

N | \(\varDelta t=0.1\varDelta x\) | \(\varDelta t=1.0\varDelta x\) | ||||||
---|---|---|---|---|---|---|---|---|

\(L^2 \) error | Order | \( L^\infty \) error | Order | \(L^2 \) error | Order | \( L^\infty \) error | Order | |

8 | 2.35E–02 | – | 6.06E–02 | – | 1.29E–01 | – | 2.98E–01 | – |

16 | 9.30E–03 | 1.33 | 2.48E–02 | 1.28 | 6.95E–02 | 0.87 | 1.85E–01 | 0.69 |

32 | 4.22E–03 | 1.13 | 1.12E–02 | 1.14 | 3.52E–02 | 0.98 | 9.89E–02 | 0.90 |

64 | 2.00E–03 | 1.07 | 5.33E–03 | 1.07 | 1.54E–02 | 1.19 | 4.35E–02 | 1.18 |

\(\varDelta t=5.0\varDelta x\) | \(\varDelta t=10.0\varDelta x\) | |||||||
---|---|---|---|---|---|---|---|---|

\(L^2 \) error | Order | \( L^\infty \) error | Order | \(L^2 \) error | Order | \( L^\infty \) error | Order | |

8 | 2.23E–01 | – | 5.10E–01 | – | 2.23E–01 | – | 5.10E–01 | – |

16 | 2.11E–01 | – | 5.37E–01 | – | 2.11E–01 | – | 5.37E–01 | – |

32 | 1.86E–01 | 0.18 | 5.07E–01 | 0.08 | 2.07E–01 | – | 5.33E–01 | – |

64 | 9.42E–02 | 0.98 | 2.61E–01 | 0.96 | 1.85E–01 | 0.16 | 5.08E–01 | 0.07 |

The number of FAS MG iterations required to reduce the norm of the residual below the tolerance \(\epsilon =1.0 \times 10^{-9}\) for \(\mathcal{P }^1\) approximation in three space dimension

\(\varDelta x\) | Jacobi smoother | Gauss–Seidel smoother |
---|---|---|

\(2\pi /16\) | 13 | 9 |

\(2\pi /32\) | 14 | 9 |

\(2\pi /64\) | 15 | 9 |

*Example 9*

ARK and DIRK methods for constant mobility

## 6 Concluding Remarks

In this paper, we have presented the linear MG solver and the nonlinear FAS MG solver for the linear and nonlinear algebraic systems arising from the LDG spatial discretization and implicit time marching methods. We have studied the CH equation in one, two and three dimensions with a constant mobility and degenerate mobility. The unconditional stability of the discrete energy is proved for the CH equation with a special homogeneous free energy density \(\varPsi (u)=\frac{1}{4}(1-u^2)^2\) based on the convex splitting method. The convergence of the linearization scheme is studied numerically by the Local Mode Analysis. Numerical experiments show that the ARK method and the DIRK method are efficient implicit time marching methods comparing to the explicit method for the CH equation. In addition, the linear MG solver and the nonlinear FAS MG solver are efficient solvers that can solve the linear and nonlinear algebraic systems with the number of iterations independent of the problem size. The optimal or sub-optimal complexity of the MG solver for \(\mathcal{P }^1\) and \(\mathcal{P }^2\) approximation are numerically shown. For \(\mathcal{P }^2\) approximation, we see the complexity of the MG solver is not always optimal. The main possible reason is that the condition number of the discretization matrix is extremely large for high order spatial discretization. Therefore, it is necessary to introduce a preconditioner such that the discretization matrix is well conditioned. We leave this topic to our future work.

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