Journal of Scientific Computing

, Volume 58, Issue 2, pp 380–408 | Cite as

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

Article

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

The Cahn–Hilliard (CH) equation in \(\varOmega \in \mathbb R ^{d}(d\le 3)\) is
$$\begin{aligned} \left\{ \begin{array}{c} u_t=\nabla \cdot (b(u)\nabla w),\\ w=-\gamma \varDelta u+\varPsi ^{\prime }(u),\\ \end{array} \right. \end{aligned}$$
(1.1)
with the boundary condition
$$\begin{aligned} \frac{\partial u}{\partial \nu }=b(u) \frac{\partial w}{\partial \nu }=0, \quad \mathrm {on} \quad \partial \varOmega , \end{aligned}$$
(1.2)
where \(\gamma \) is a positive constant. The mobility \(b(u)\) is non-negative and the function \(\varPsi (u)\) represents the homogeneous free energy in the energy functional
$$\begin{aligned} \varepsilon (u)= \int \limits _\varOmega (\frac{\gamma }{2}|\nabla u|^2+ \varPsi (u))dx. \end{aligned}$$
For the fourth order nonlinear CH equation, explicit time discretization methods will require severe time step restriction \((\varDelta t \sim O(\varDelta x^4))\), so we explore implicit methods. Being implicit in time, it requires to solve the system of algebraic equations at each time step, arising by the local discontinuous Galerkin (LDG) spatial discretization and high order time discretization methods. The efficiency of the implicit methods highly depends on the efficiency of the solver.

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.

Let \(\mathcal{P }^k(K)\) be the space of polynomials of degree at most \(k\ge 0\) on \(K\). The finite element spaces associated with the mesh are of the form
$$\begin{aligned} V_h^k&= \{v \in L^2(\varOmega ): v|_K \in \mathcal{P }^k(K), \forall K \in \mathcal{T }_h\},\\ \varSigma _h^k&= \{{\mathbf{w}}=(w_1,\ldots ,w_d)^T \in L^2(\varOmega )^d: w_l|_K \in \mathcal{P }^k(K), l=1,\ldots ,d, \forall K \in \mathcal{T }_h \} \end{aligned}$$
Further, we define the inner product notations as
$$\begin{aligned} (w,v)_K&= \int \limits _K wv dK, \quad (w,v)_{\partial K}=\int \limits _{\partial K} wv ds, \end{aligned}$$
(2.1)
$$\begin{aligned} ({\mathbf{q}},{\mathbf{p}})_K&= \int \limits _K {\mathbf{q}} \cdot {\mathbf{p}}dK, \quad ({\mathbf{q}},{\mathbf{p}})_{\partial K}=\int \limits _{\partial K} {\mathbf{q}} \cdot {\mathbf{p}}ds, \end{aligned}$$
(2.2)
for the scalar variables \(w, v\) and vector variables \({\mathbf{q}}, {\mathbf{p}}\) respectively. The inner products on \(\varOmega \) are defined as
$$\begin{aligned} (w,v)_\varOmega =\sum _K(w,v)_K, \quad ({\mathbf{q}},{\mathbf{p}})_\varOmega =\sum _K ({\mathbf{q}},{\mathbf{p}})_K. \end{aligned}$$
(2.3)

2.2 Full-Discrete LDG Scheme

Wise [33] introduced a convex splitting finite difference scheme for the Cahn–Hilliard–Hele–Shaw (CHHS) equation and proved the unconditional energy stability. In this subsection, we apply the convex splitting method to CH equation coupled with the LDG spatial discretization for a special choice of \(\varPsi (u)\). We refer the readers to [34] for the general LDG scheme. The homogeneous free energy density \(\varPsi (u)\) is taken as \(\frac{1}{4}(1-u^2)^2\) and the full-discrete convex splitting scheme is
$$\begin{aligned} \frac{u^{n+1}-u^n}{\varDelta t}=\nabla \cdot (b(u^{n})\nabla (-\gamma \varDelta u^{n+1}+({u^{n+1}})^3-u^n )). \end{aligned}$$
(2.4)
In order to define the LDG method to the equation (2.4), we first rewrite it as a first order system:
$$\begin{aligned} \frac{u^{n+1}-u^n}{\varDelta t}&= \nabla \cdot {{\mathbf{s}}^{n+1}}, \end{aligned}$$
(2.5a)
$$\begin{aligned} {{\mathbf{s}}^{n+1}}&= b(u^{n}){{\mathbf{p}}^{n+1}},\end{aligned}$$
(2.5b)
$$\begin{aligned} {{\mathbf{p}}^{n+1}}&= \nabla (r^{n+1}-q^{n+1}),\end{aligned}$$
(2.5c)
$$\begin{aligned} q^{n+1}&= \gamma \nabla \cdot {\mathbf{w}}^{n+1}, \end{aligned}$$
(2.5d)
$$\begin{aligned} {\mathbf{w}}^{n+1}&= \nabla u^{n+1},\end{aligned}$$
(2.5e)
$$\begin{aligned} r^{n+1}&= ({u^{n+1}})^3-u^n. \end{aligned}$$
(2.5f)
To simplify the notation, we still use \(u^{n+1},{{\mathbf{s}}^{n+1}},{{\mathbf{p}}^{n+1}},q^{n+1},{\mathbf{w}}^{n+1}\) and \(r^{n+1}\) to denote the numerical solutions. The LDG scheme to solve the system (2.5) is as follows: Find \((u^{n+1}, {{\mathbf{s}}^{n+1}},{{\mathbf{p}}^{n+1}},q^{n+1},{\mathbf{w}^{n+1}},r^{n+1}) \in V_h^k \times \varSigma _h^k \times \varSigma _h^k \times V_h^k \times \varSigma _h^k \times V_h^k\) such that, \( \forall \rho ,{\varvec{\theta }},{\varvec{\eta }},\varphi , {{\varvec{\phi }}},\xi \in V_h^k \times \varSigma _h^k \times \varSigma _h^k \times V_h^k \times \varSigma _h^k \times V_h^k\)
$$\begin{aligned} (\frac{u^{n+1}-u^n}{\varDelta t}, \rho )_K&= -({{\mathbf{s}}^{n+1}} , \nabla \rho )_K+(\widehat{\mathbf{s}^{n+1} \cdot {\varvec{\nu }}} ,\rho )_{\partial K}, \end{aligned}$$
(2.6a)
$$\begin{aligned} (\mathbf{s}^{n+1} ,{\varvec{\theta }})_K&= ( b(u^{n})\mathbf{p}^{n+1} , {\varvec{\theta }})_K , \end{aligned}$$
(2.6b)
$$\begin{aligned} ({{\mathbf{p}}^{n+1}},{\varvec{\eta }} )_K&= -(r^{n+1}-q^{n+1}, \nabla \cdot {\varvec{\eta }} )_K + (\widehat{r}^{n+1}-\widehat{q}^{n+1}, {\varvec{\nu }} \cdot {\varvec{\eta }})_{\partial K}, \end{aligned}$$
(2.6c)
$$\begin{aligned} (q^{n+1} ,\varphi )_K&= -\gamma ({\mathbf{w}}^{n+1} , \nabla \varphi )_K+\gamma (\widehat{{\mathbf{w}}^{n+1} \cdot {\varvec{\nu }}} ,\varphi )_{\partial K}, \end{aligned}$$
(2.6d)
$$\begin{aligned} ({\mathbf{w}}^{n+1} , {\varvec{\phi }})_K&= -( u^{n+1}, \nabla \cdot {\varvec{\phi }})_K +( \widehat{u}^{n+1} ,{\varvec{\phi }} \cdot {\varvec{\nu }})_{\partial K}, \end{aligned}$$
(2.6e)
$$\begin{aligned} ( r^{n+1} ,\xi )_K&= (({u^{n+1}})^3-u^n ,\xi )_K. \end{aligned}$$
(2.6f)
The “hat” terms in (2.6) at the cell boundary from integration by parts are the so-called “numerical fluxes”, which are functions defined on the edges and should be designed based on different guiding principles for different PDEs to ensure stability. It turns out that we can take the simple choices such that
$$\begin{aligned} \widehat{\mathbf{s}}^{n+1}|_e= \widehat{\mathbf{s}}^{n+1}_L, \widehat{q}^{n+1}|_e=\widehat{q}^{n+1}_R, \widehat{r}^{n+1}|_e=\widehat{r}^{n+1}_R, \widehat{\mathbf{w}}^{n+1}|_e= \widehat{\mathbf{w}}^{n+1}_L, \widehat{u}^{n+1}|_e=\widehat{u}^{n+1}_R.\qquad \end{aligned}$$
(2.7)
According to the boundary condition (1.2), we take
$$\begin{aligned} \widehat{{\mathbf{s}}}^{n+1}= {\mathbf{0}},\widehat{\mathbf{w}}^{n+1}= {\mathbf{0}}, \widehat{u}^{n+1}=({u^{n+1}})^\mathrm in ,\widehat{q}^{n+1} =({q^{n+1}})^\mathrm in , \widehat{r}^{n+1}=({r^{n+1}})^\mathrm in , \end{aligned}$$
(2.8)
at the domain boundary, where \(({u^{n+1}})^\mathrm in \) means the value taking from the inside of the boundary element.
We choose a local basis in cell \(K\), then \(\mathbf{s}^{n+1}, \mathbf{p}^{n+1}, {\mathbf{w}}^{n+1}\) and \(r^{n+1}\) can be eliminated from equations (2.6b), (2.6c), (2.6e) and (2.6a), respectively, by simply inverting a small mass matrix in each case. Then we obtain a system of two coupled equations for \([u^{n+1},q^{n+1}]\)
$$\begin{aligned} GU=F, \end{aligned}$$
(2.9)
where \(U=[u^{n+1}, q^{n+1}]^{T}\) and \(F\) is the corresponding right hand side vector consisting of \(u^n\) and \(q^n\).

2.3 Energy Stability

In this subsection, we will prove the energy stability for the full-discrete LDG scheme based on the convex splitting method. To simplify the notation, we use the following notations for discretization of time variable,
$$\begin{aligned} \delta _t u^{n+1}&= \frac{u^{n+1}-u^n}{\varDelta t},\\ \delta _t {\mathbf{w}}^{n+1}&= \frac{{\mathbf{w}}^{n+1}-{\mathbf{w}}^n}{\varDelta t}. \end{aligned}$$

Proposition 1

(Energy stability) The solution to the scheme (2.6) and the flux (2.7) satisfies the energy stability
$$\begin{aligned} \frac{1}{\varDelta t}[\frac{\gamma }{2}({\mathbf{w}}^{n+1},{\mathbf{w}}^{n+1})_\varOmega&+ (\varPsi (u^{n+1}),1)_\varOmega ]+(b(u^{n}){\mathbf{p}}^{n+1}, {\mathbf{p}}^{n+1})_\varOmega \\&\le \frac{1}{\varDelta t} [\frac{\gamma }{2}({\mathbf{w}}^{n},{\mathbf{w}}^{n})_\varOmega +(\varPsi (u^{n}),1 )_\varOmega ]. \end{aligned}$$

Proof

We take the test functions in (2.6) as
$$\begin{aligned} \rho&= r^{n+1}-q^{n+1},\quad {\varvec{\theta }}=-{\mathbf{p}}^{n+1}, \quad {\varvec{\eta }}={\mathbf{s}}^{n+1},\nonumber \\ \varphi&= \delta _t u^{n+1},\quad {\varvec{\phi }}=\gamma \delta _t {\mathbf{w}}^{n+1},\quad \xi =-\delta _t u^{n+1}. \end{aligned}$$
(2.10)
Then we have
$$\begin{aligned} (\delta _t u^{n+1}, r^{n+1}-q^{n+1})_K&= -({{\mathbf{s}}^{n+1}} , \nabla (r^{n+1}-q^{n+1}))_K +( \widehat{\mathbf{s}^{n+1} \cdot {\varvec{\nu }}} ,r^{n+1}-q^{n+1})_{\partial K}, \nonumber \\ -( \mathbf{s}^{n+1}, {\mathbf{p}}^{n+1})_K&= -( b(u^{n})\mathbf{p}^{n+1}, {\mathbf{p}}^{n+1})_K, \nonumber \\ ({{\mathbf{p}}^{n+1}} , {\mathbf{s}}^{n+1})_K&= -(r^{n+1}-q^{n+1} ,\nabla \cdot {\mathbf{s}}^{n+1})_K + (\widehat{r}^{n+1}-\widehat{q}^{n+1} ,{\varvec{\nu }} \cdot {\mathbf{s}}^{n+1})_{\partial K}, \nonumber \\ (q^{n+1} ,\delta _t u^{n+1} )_K&= -\gamma ( {\mathbf{w}}^{n+1} , \nabla \delta _t u^{n+1})_K+\gamma (\widehat{{\mathbf{w}}^{n+1} \cdot {\varvec{\nu }}} ,\delta _t u^{n+1})_{\partial K}, \nonumber \\ \gamma ({\mathbf{w}}^{n+1}, \delta _t {\mathbf{w}}^{n+1} )_K&= -\gamma (u^{n+1}, \nabla \cdot \delta _t {\mathbf{w}}^{n+1})_K +\gamma (\widehat{u}^{n+1}, \delta _t {\mathbf{w}}^{n+1} \cdot {\varvec{\nu }})_{\partial K}, \nonumber \\ -(r^{n+1}, \delta _t u^{n+1} )_K&= -(({u^{n+1}})^3-u^n ,\delta _t u^{n+1})_K. \end{aligned}$$
(2.11)
Summing up equations in (2.11), we obtain
$$\begin{aligned}&(\gamma {\mathbf{w}}^{n+1} , \delta _t {\mathbf{w}}^{n+1})_K+((({u^{n+1}})^3-u^n)\delta _t u^{n+1},1)_K+(b(u^n){\mathbf{p}}^{n+1}, {\mathbf{p}}^{n+1} )_K \\&=-(r^{n+1}-q^{n+1},{\mathbf{s}}^{n+1} \cdot {\varvec{\nu }})_{\partial K}+(\widehat{\mathbf{s}^{n+1}\cdot {\varvec{\nu }}},r^{n+1}-q^{n+1})_{\partial K} +(\widehat{r}^{n+1}-\widehat{q}^{n+1},{\mathbf{s}}^{n+1} \cdot {\varvec{\nu }})_{\partial K}\\&\quad +\frac{\gamma }{\varDelta t}[-( u^{n+1},{\mathbf{w}}^{n+1}\cdot {\varvec{\nu }})_{\partial K}+(\widehat{{\mathbf{w}}^{n+1}\cdot {\varvec{\nu }}},u^{n+1})_{\partial K}+( \widehat{u}^{n+1},{\mathbf{w}}^{n+1}\cdot {\varvec{\nu }})_{\partial K}] \\&\quad +\frac{\gamma }{\varDelta t} [({\mathbf{w}}^{n+1}, \nabla u^{n})_K+( u^{n+1},\nabla \cdot {\mathbf{w}}^n)_K - (\widehat{{\mathbf{w}}^{n+1}\cdot {\varvec{\nu }}},u^{n})_{\partial K}-( \widehat{u}^{n+1},{\mathbf{w}}^n\cdot {\varvec{\nu }})_{\partial K}]. \end{aligned}$$
In the above analysis, we choose the test functions as \(\delta _t u^{n+1}, \delta _t {\mathbf{w}}^{n+1}\), thus the cross (the time level \(n\) and \(n+1\)) terms \(({\mathbf{w}}^{n+1},\nabla u^{n})_K, (u^{n+1},\nabla \cdot {\mathbf{w}}^n)_K\) occur. What we need to do next is to eliminate the cross terms. For equation (2.6e), we choose the test function as \({\varvec{\phi }}={\mathbf{w}}^n\), then we obtain
$$\begin{aligned} ({\mathbf{w}}^{n+1} , {\mathbf{w}}^n)_K = -( u^{n+1}, \nabla \cdot {\mathbf{w}}^n)_K +( \widehat{u}^{n+1} ,{\mathbf{w}}^n \cdot {\varvec{\nu }})_{\partial K}. \end{aligned}$$
(2.12)
From (2.6e), we have
$$\begin{aligned} ({\mathbf{w}}^{n} , {\varvec{\phi }})_K = -( u^{n}, \nabla \cdot {\varvec{\phi }})_K +( \widehat{u}^{n} ,{\varvec{\phi }} \cdot {\varvec{\nu }})_{\partial K}. \end{aligned}$$
(2.13)
Choosing the test function \({\varvec{\phi }}={\mathbf{w}}^{n+1}\) in (2.13), we obtain
$$\begin{aligned} ({\mathbf{w}}^{n} , {\mathbf{w}}^{n+1})_K = -( u^{n}, \nabla \cdot {\mathbf{w}}^{n+1})_K +( \widehat{u}^{n} ,{\mathbf{w}}^{n+1} \cdot {\varvec{\nu }})_{\partial K}. \end{aligned}$$
(2.14)
Then we have
$$\begin{aligned}&({\mathbf{w}}^{n+1},\nabla u^{n})_K+(u^{n+1},\nabla \cdot {\mathbf{w}}^n)_K - (\widehat{{\mathbf{w}}^{n+1}\cdot {\varvec{\nu }},u^{n}})_{\partial K}-(\widehat{u}^{n+1},{\mathbf{w}}^n\cdot {\varvec{\nu }})_{\partial K}\\&\,\stackrel{(2.12)}{=} -({\mathbf{w}}^{n+1}, {\mathbf{w}}^{n})_K+({\mathbf{w}}^{n+1}, \nabla u^{n})_K - (\widehat{{\mathbf{w}}^{n+1}\cdot {\varvec{\nu }}},u^{n})_{\partial K}\\&\quad {=}-({\mathbf{w}}^{n+1},{\mathbf{w}}^{n})_K+({\mathbf{w}}^{n+1}, \nabla u^{n})_K-(\widehat{{\mathbf{w}}^{n+1}\cdot {\varvec{\nu }}},u^{n})_{\partial K} + (u^n, \nabla \cdot {\mathbf{w}}^{n+1})_K\\&\qquad -(\widehat{u}^n,{\mathbf{w}}^{n+1}\cdot {\varvec{\nu }})_{\partial K}- (u^n,\nabla \cdot {\mathbf{w}}^{n+1})_K+(\widehat{u}^n,{\mathbf{w}}^{n+1}\cdot {\varvec{\nu }})_{\partial K}\\&\,\stackrel{(2.14)}{=}({\mathbf{w}}^{n+1}, \nabla u^{n})_K-(\widehat{{\mathbf{w}}^{n+1}\cdot {\varvec{\nu }}},u^{n})_{\partial K} + (u^n, \nabla \cdot {\mathbf{w}}^{n+1})_K -(\widehat{u}^n,{\mathbf{w}}^{n+1}\cdot {\varvec{\nu }})_{\partial K}. \end{aligned}$$
According to the above relations, we have
$$\begin{aligned}&(\gamma {\mathbf{w}}^{n+1}, \delta _t {\mathbf{w}}^{n+1})_K+((({u^{n+1}})^3-u^n)\delta _t u^{n+1},1)_K+(b(u^n){\mathbf{p}}^{n+1}, {\mathbf{p}}^{n+1} )_K \\&=-(r^{n+1}-q^{n+1},{\mathbf{s}}^{n+1} \cdot {\varvec{\nu }})_{\partial K}+(\widehat{s^{n+1} \cdot {\varvec{\nu }}},r^{n+1}-q^{n+1})_{\partial K} +(\widehat{r}^{n+1}-\widehat{q}^{n+1}, {\mathbf{s}}^{n+1} \cdot {\varvec{\nu }})_{\partial K}\\&\quad +\frac{\gamma }{\varDelta t}[-(u^{n+1},{\mathbf{w}}^{n+1}\cdot {\varvec{\nu }})_{\partial K}+ (\widehat{{\mathbf{w}}^{n+1}\cdot {\varvec{\nu }}},u^{n+1})_{\partial K}+(\widehat{u}^{n+1},{\mathbf{w}}^{n+1}\cdot {\varvec{\nu }})_{\partial K}]\\&\quad +\frac{\gamma }{\varDelta t}[({\mathbf{w}}^{n+1}, \nabla u^{n})_K + (u^n, \nabla \cdot {\mathbf{w}}^{n+1})_K -(\widehat{u}^n,{\mathbf{w}}^{n+1}\cdot {\varvec{\nu }})_{\partial K}-(\widehat{{\mathbf{w}}^{n+1}\cdot {\varvec{\nu }}},u^{n})_{\partial K}]. \end{aligned}$$
Summing up over \(K\), with the numerical fluxes (2.7) and the above specific choices of the fluxes (2.8) at the domain boundary, we get
$$\begin{aligned}&(\gamma {\mathbf{w}}^{n+1} , \delta _t {\mathbf{w}}^{n+1})_\varOmega +((({u^{n+1}})^3-u^n)\delta _t u^{n+1},1)_\varOmega +(b(u^n){\mathbf{p}}^{n+1}, {\mathbf{p}}^{n+1} )_\varOmega \\&\quad =(\gamma {\mathbf{w}}^{n+1}, \delta _t {\mathbf{w}}^{n+1})_\varOmega +((({u^{n+1}})^3-u^n) \delta _t u^{n+1},1)_\varOmega -\frac{1}{4\varDelta t}((1-({u^{n+1}})^2)^2,1)_\varOmega \\&\qquad +\frac{1}{4\varDelta t}((1-({u^{n}})^2)^2,1)_\varOmega +\frac{1}{4\varDelta t}((1-({u^{n+1}})^2)^2,1)_\varOmega -\frac{1}{4\varDelta t}((1-({u^{n}})^2)^2,1)_\varOmega \\&\qquad +(b(u^n){\mathbf{p}}^{n+1}, {\mathbf{p}}^{n+1} )_\varOmega \\&\quad =(\gamma {\mathbf{w}}^{n+1}, \delta _t {\mathbf{w}}^{n+1})_\varOmega +\frac{1}{4\varDelta t}((1-({u^{n+1}})^2)^2,1)_\varOmega -\frac{1}{4\varDelta t}((1-({u^{n}})^2)^2,1)_\varOmega \\&\qquad +(b(u^{n}){\mathbf{p}}^{n+1}, {\mathbf{p}}^{n+1})_\varOmega +\frac{1}{4 \varDelta t}((2+2({u^{n+1}})^2+(u^n+u^{n+1})^2(u^n-u^{n+1})^2),1)_\varOmega \\&\quad = 0. \end{aligned}$$
Then we get
$$\begin{aligned} (\gamma {\mathbf{w}}^{n+1}, \delta _t {\mathbf{w}}^{n+1})_\varOmega +(b(u^{n}){\mathbf{p}}^{n+1}, {\mathbf{p}}^{n+1})_\varOmega \!+\!\frac{1}{4\varDelta t}((1\!-\!({u^{n+1}})^2)^2\!-\!(1-({u^{n}})^2)^2,1)_\varOmega \!\le \!0, \end{aligned}$$
so we have the energy stability
$$\begin{aligned}&\frac{1}{\varDelta t}[\frac{\gamma }{2}({\mathbf{w}}^{n+1},{\mathbf{w}}^{n+1})_\varOmega +(\varPsi (u^{n+1}),1)_\varOmega ]+(b(u^{n}){\mathbf{p}}^{n+1},{\mathbf{p}}^{n+1})_\varOmega \\&\le \frac{1}{\varDelta t} [\frac{\gamma }{2}({\mathbf{w}}^{n},{\mathbf{w}}^{n})_\varOmega +(\varPsi (u^{n}),1 )_\varOmega ]. \end{aligned}$$
\(\square \)

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.

Starting from \(u^n\) (the numerical approximation of \(u(t^n)\), with \(u^0=u(0)\)), we give the method to calculate \(u^{n+1}\) in the following.
  1. 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. 2.
    Apply the LDG spatial discretization similar to [34] to the equation (3.1) and obtain an ODE
    $$\begin{aligned} {\left\{ \begin{array}{ll} u_t=F_N(t,u(t))+F_S(t,u(t)), &{}t \in [0,T],\\ u(0)=u^0, &{} \end{array}\right. } \end{aligned}$$
    (3.2)
    where \(F_S(t,u(t))\) 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).
     
  3. 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. 4.

    Apply the linear MG method to solve the linear system.

     

3.2 The Nonlinear Scheme

For CH equation (1.1), especially for the equation with the degenerate mobility \(b(u)\), we can also consider a full implicit scheme instead of the linearization technique. We first introduce the DIRK method, here we just take the third order DIRK method for an example, which is used in our numerical experiments. We apply the LDG spatial discretization [34] to the CH equation (1.1) and get an ODE
$$\begin{aligned} {\left\{ \begin{array}{ll} u_t=F(t,u(t)), &{}t \in [0,T],\\ u(0)=u_0, &{} \end{array}\right. } \end{aligned}$$
(3.3)
where \(F(t,u(t))\) is a nonlinear function of \(u\).
Starting from \(u^n\), we give the DIRK method to calculate \(u^{n+1}\). Following the work [1], the DIRK method is used to solve equation (3.3). They are given in the following form
$$\begin{aligned} u^{n,i}&= u^n+ \varDelta t \sum _{j=1}^3 a_{ij}F(t^{n,j},u^{n,j}), \quad i=1,2,3,\\ u^{n+1}&= u^n+ \varDelta t \sum _{j=1}^3 b_jF(t^{n,j},u^{n,j}),\nonumber \end{aligned}$$
(3.4)
where \(t^{n,j}=t^n+c_j \varDelta t\) and \(u^{n,j}\) approximates \(u(t^{n,j})\). The coefficients \(a_{ij},i=1,2,3, b_j, c_j\) are constrained by order of accuracy and stability considerations. For a detailed description of the method, we refer the readers to [1].

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

We start by introducing the basic notations to describe the general setting of a two-level or bi-grid method. Together with the family of partitions \(\{\mathcal{T }_h\}_{h>0}\) used for the LDG discretization, we consider a coarse family of mesh partitions, \(\{\mathcal{T }_H\}_{H>0}\) with \(H>h\) and satisfying the basic assumption \(\mathcal{T }_H \subset \mathcal{T }_h\). One can think \(H=2h\), since in many circumstances it will already be coarse enough. Associated to the coarse mesh partition we have the corresponding finite element space \(V_H^{k}\) which is defined as:
$$\begin{aligned} V_H^{k}=\{v \in \, L^{2}(D): v|_D\in \mathcal{P }^k(D);\quad \forall \, D\in \mathcal{T }_H\}. \end{aligned}$$
(4.1)
Throughout the whole description we assume the polynomial degree \(k\) is fixed.
To link functions in both spaces, we define the prolongation and restriction operators. The prolongation operator \(P_{hH}:V_H \vee V_h\) is defined as the natural inclusion. The restriction operator \(R_{Hh} : V_h \vee V_H\) is defined as the transpose of \(P_{hH}\) with respect to the standard \(L^{2}\)-inner product. That is, it is obtained by solving:
$$\begin{aligned} \sum _{D\in \mathcal{T }_H}\int \limits _{D} R_{Hh}(u_h) v_{H} dx=\sum _{D\in \mathcal{T }_h} \int \limits _{D} u_h P_{hH}(v_{H}) dx,\quad \forall v_{H} \in V_H^{k}. \end{aligned}$$
(4.2)
We now denote \(S_h\) as a general relaxation or smoothing operator. We specify and study several choices in “Appendix 1”. The basic property that \(S_h\) should have is to damp the high frequencies of the approximate solution and smooth the error. The coarse solver is defined by \(A_{H}= R_{Hh}A_hP_{hH}\). The coarse grid correction step is to reduce the smooth components of the error that can not be reduced by the smoother.

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

In this section, we consider the Local Mode Analysis for the linearization scheme (3.1). The linear part is \(\nabla \cdot (b(u^n)\nabla (-\gamma \varDelta u))\), so what we do in the following is to analyze to the equation
$$\begin{aligned} \frac{u^{n+1}-u^n}{\varDelta t}=\nabla \cdot (b(u^n)\nabla (-\gamma \varDelta u^{n+1})). \end{aligned}$$
(4.3)
The analysis for general fourth order linear equation can be found in [21]. For convenience we consider the Local Mode Analysis for the one-dimensional and two-dimensional cases. Extension to higher dimensions follows immediately by means of the tensor-product principle. For a detailed description of the Local Mode Analysis for the two-grid algorithm, we refer the readers to “Appendix 3”.
We choose \(b(u)=1, \gamma =1\) in Eq. (4.3). The convergence of the two-grid algorithm in one-dimensional and two-dimensional cases are analyzed by the Local Mode Analysis and the numerical results are shown as follows.
  • One space dimension
    1. 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. 2)

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

       
    3. 3)

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

       
  • Two space dimension
    1. 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. 2)

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

       
    3. 3)

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

       
Fig. 1

The asymptotic convergence factor \(\lambda \) changes with the damping parameter \(\alpha \) in one dimension

Fig. 2

Eigenvalue spectral of \(E_h\) and \(E_h^{2grid}\) with damped Jacobi smoother and \(\mathcal{P }^1\) approximation, \(\mathcal{P }^2\) approximation in one dimension

Fig. 3

Eigenvalue spectral of \(E_h\) and \(E_h^{2grid}\) with Gauss–Seidel smoother and \(\mathcal{P }^1\) approximation, \(\mathcal{P }^2\) approximation in one dimension

Fig. 4

The asymptotic convergence factor \(\lambda \) changes with the damping parameter \(\alpha \) in two dimension

Fig. 5

Eigenvalue spectral of \(E_h\) and \(E_h^{2grid}\) with damped Jacobi smoother and \(\mathcal{P }^1\) approximation, \(\mathcal{P }^2\) approximation in two dimension

Fig. 6

Eigenvalue spectral of \(E_h\) and \(E_h^{2grid}\) with Gauss–Seidel smoother and \(\mathcal{P }^1\) approximation, \(\mathcal{P }^2\) approximation in two dimension

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

We consider the scheme (2.4) in \(\varOmega =(0,2\pi )\) with degenerate mobility \(b(u)=1-u^2, \gamma =2, \varPsi (u)=\frac{1}{4}(1-u^2)^2\) and periodic boundary conditions. We take the exact solution of
$$\begin{aligned} u(x,t)=e^{-t}\sin (x), \end{aligned}$$
(5.1)
with the source term \(f\), where \(f\) is a given function so that (5.1) is the exact solution.
The convex splitting scheme (2.4) is the first order accuracy in time, so we just consider \(\mathcal{P }^0\) approximation. The \(L^2\) and \(L^\infty \) errors, and the numerical order of accuracy at time \(t=1.0\) with different time steps are presented in Table 1. The numerical experiments go well with the theoretical result of the unconditional energy stability for the scheme.
Table 1

Accuracy test for the scheme (2.4) with the exact solution (5.1)

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

We consider the CH equation
$$\begin{aligned} u_t=(b(u)(-\gamma u_{xxx}+(\varPsi ^{\prime }(u))_x))_x, \end{aligned}$$
(5.2)
with \(b(u)=1, \varPsi (u)=\frac{3}{2}(1-u^2),\gamma =4\) in \(\varOmega =(0,4\pi )\) and periodic boundary conditions. The exact solution is given by
$$\begin{aligned} u(x,t)=e^{-t}\sin (x)-e^{0.5t}\sin (0.5x). \end{aligned}$$
(5.3)
The implicit third order ARK method is used in the time integration and the time step is taken as \(\varDelta t=0.1 \varDelta x\). The \(L^2\) and \(L^\infty \) errors, and the numerical order of accuracy at time \(t=0.5\) are presented in Table 2. We can see that the method with \(\mathcal{P }^k\) elements gives a \((k+1)\)-th order of accuracy.
Table 2

Accuracy test for the CH equation (5.2) with the exact solution (5.3)

 

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

To illustrate the superiority of the MG solver, we present the convergence rates of the method for a single time-step by using \(\mathcal{P }^1\) and \(\mathcal{P }^2\) elements. From Fig. 7, we can see that each iteration of the MG solver is an \(O(N)\) operation. By Table 3 we see that the convergence is independent of the mesh size. We also find that the Gauss–Seidel smoother show better convergence than the Jacobi smoother. The same information for \(\mathcal{P }^2\) approximation are shown in Table 4 and Fig. 8 and we conclude that \(\mathcal{P }^2\) approximation converges worse than \(\mathcal{P }^1\) approximation.
Fig. 7

Convergence rates of MG solver for \(\mathcal{P }^1\) approximation in one space dimension

Fig. 8

Convergence rates of MG solver for \(\mathcal{P }^2\) approximation with Gauss-Seidel smoother in one space dimension

Table 3

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

Table 4

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

We consider the CH equation
$$\begin{aligned} u_t=(b(u)(-\gamma u_{xxx}+(\varPsi ^{\prime }(u))_x))_x-f(x,t), \end{aligned}$$
(5.4)
with \(b(u)=1, \varPsi (u)=\frac{1}{4}(1-u^2)^2,\gamma =2\) in \(\varOmega =(0,4\pi )\) and periodic boundary conditions. We take the exact solution
$$\begin{aligned} u(x,t)=e^{-t}\sin (x)-e^{0.5t}\cos (0.5x). \end{aligned}$$
(5.5)
The implicit third order ARK method and the third order DIRK method are used in the time integration with the time step \(\varDelta t=0.05 \varDelta x\) and \(\varDelta t=0.1 \varDelta x\) respectively. The \(L^2\) and \(L^\infty \) errors, and the numerical order of accuracy at time \(t=0.5\) are presented in Table 5. We can see that both methods with \(\mathcal{P }^k\) elements give a \((k+1)\)-th order of accuracy. The CPU time with these two different time discretization methods are shown in Fig. 9. We can see that the ARK method with linear MG solver cost less CPU time. The reason is that for constant mobility \(b(u)\), we do not need the linearization technique, so the ARK method is more efficient.
Fig. 9

The CPU time of the linear scheme and the nonlinear scheme for the CH equation (5.4) with \(\mathcal{P }^1\) approximation

Table 5

Accuracy test for the CH equation (5.4) with the exact solution (5.5)

 

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

We consider the CH equation
$$\begin{aligned} u_t=(b(u)(-\gamma u_{xxx}+(\varPsi ^{\prime }(u))_x))_x-f(x,t), \end{aligned}$$
(5.6)
with \(b(u)=1-u^2, \varPsi (u)=\frac{1}{2}(1-u^2),\gamma =2\) in \(\varOmega =(0,2\pi )\) and periodic boundary conditions. We take the exact solution
$$\begin{aligned} u(x,t)=e^{-t}\sin (x). \end{aligned}$$
(5.7)
The implicit third order ARK method and the third order DIRK method are used in the time integration with the time step \(\varDelta t=0.001 \varDelta x\) and \(\varDelta t=0.07 \varDelta x^2\) respectively. The \(L^2\) and \(L^\infty \) errors, and the numerical order of accuracy at time \(t=0.5\) are presented in Table 6. We can see that both methods with \(\mathcal{P }^k\) elements give a \((k+1)\)-th order of accuracy. The CPU time with the two different time discretization methods are shown in Fig. 10. We can see that the DIRK method with nonlinear FAS MG solver cost less CPU time. The reason is that for degenerate mobility \(b(u)\), the linearization technique is needed and the time step can not be chosen larger because of the explicit treatment of the term \(\nabla \cdot ((b(u)-b(u^n))\nabla (-\gamma \varDelta u))\), so under this situation, the DIRK method is more efficient.
Fig. 10

The CPU time of the linear scheme and the nonlinear scheme for the CH equation (5.6) with \(\mathcal{P }^1\) approximation

Table 6

Accuracy test for the CH equation (5.6) with the exact solution (5.7)

 

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

We consider the scheme (2.4) in \(\varOmega =(0,2\pi ) \times (0,2\pi )\) with degenerate mobility \(b(u)=1-u^2, \gamma =1, \varPsi (u)=\frac{1}{4}(1-u^2)^2\) and periodic boundary conditions. We take the exact solution of
$$\begin{aligned} u(x,y,t)=e^{-2t}\sin (x)\sin (y), \end{aligned}$$
(5.8)
with the source term \(f\), where \(f\) is a given function so that (5.8) is the exact solution.
The convex splitting scheme (2.4) is the first order accuracy in time, so we just consider \(\mathcal{P }^0\) approximation. The \(L^2\) and \(L^\infty \) errors, and the numerical order of accuracy at time \(t=1.0\) with different time steps are presented in Table 7. The numerical experiments go well with the theoretical result of the unconditional energy stability for the scheme.
Table 7

Accuracy test for the scheme (2.4) with the exact solution (5.8)

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

To illustrate the superiority of the FAS MG solver, we present the convergence rates of the method for a single time-step by using \(\mathcal{P }^1\) and \(\mathcal{P }^2\) elements. From Fig. 11, we can see that each iteration of the FAS MG solver is an \(O(N)\) operation. From Table 8 we see that the convergence is independent of the mesh size. It is clear that the Gauss–Seidel smoother converges much faster than the Jacobi smoother. The same information for \(\mathcal{P }^2\) approximation are shown in Table 9 and Fig. 12 and we conclude that \(\mathcal{P }^1\) approximation converges better than \(\mathcal{P }^2\) approximation.
Fig. 11

Convergence rates of FAS MG solver for \(\mathcal{P }^1\) approximation in two space dimension

Fig. 12

Convergence rates of FAS MG solver for \(\mathcal{P }^2\) approximation with Gauss-Seidel smoother in two space dimension

Table 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 \(\mathcal{P }^1\) 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

Table 9

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

We consider the CH equation
$$\begin{aligned} u_t=\nabla \cdot (b(u)\nabla (-\gamma \varDelta u+\varPsi ^{\prime }(u))), \end{aligned}$$
(5.9)
with \(b(u)=1, \varPsi (u)=600(u \mathrm ln u+(1-u)\mathrm ln (1-u))+1800u(1-u),\gamma =1\). The initial condition is
$$\begin{aligned} u_0(x)= {\left\{ \begin{array}{ll} 0.71, &{} {\mathbf{x}} \in \varOmega _1,\\ 0.69, &{} {\mathbf{x}} \in \varOmega _2, \end{array}\right. } \end{aligned}$$
(5.10)
where the square domain
$$\begin{aligned} \varOmega =(-0.5,0.5)\times (-0.5,0.5), \quad \varOmega _1=(-0.2,0.2)\times (-0.2,0.2), \quad \varOmega _2=\Omega -\varOmega _1. \end{aligned}$$
The boundary conditions are
$$\begin{aligned} \frac{\partial u}{\partial \nu }=b(u)\nabla w \cdot \nu =0, \quad \mathrm {on} \quad \partial \varOmega . \end{aligned}$$
(5.11)
The implicit third order ARK method and the DIRK method are used in the time integration. We use the \(\mathcal{P }^1\) and \(\mathcal{P }^2\) elements on the meshes with \(128 \times 128\) cells, respectively. The time step is taken as \(\varDelta t=0.01 \varDelta x^2\). The contours at \(t=8 \times 10^{-5}\) are shown in Fig. 13. We can see that these two methods show the same phenomenon and the numerical results compare very well with the numerical calculations in [34].
Fig. 13

The contours of \(u({\mathbf{x}},t)\) for equation (5.9), \(\mathcal{P }^1\) and \(\mathcal{P }^2\) elements on the uniform mesh with \(128 \times 128\) cells

Example 7

ARK and DIRK methods for degenerate mobility

We consider the CH equation
$$\begin{aligned} u_t=\nabla \cdot (b(u)\nabla (-\gamma \varDelta u+\varPsi ^{\prime }(u))), \end{aligned}$$
(5.12)
with \(b(u)=u(1-u), \varPsi (u)=3000(u \mathrm ln u+(1-u)\mathrm ln (1-u))+9000u(1-u),\gamma =1\) in \(\varOmega =(-0.5,0.5)\times (-0.5,0.5)\). The boundary conditions are taken as (5.11). The initial condition \(u_0\) is a random perturbation of uniform state \(u=0.63\) with a fluctuation no larger than 0.05.
The implicit third order ARK method and the DIRK method are used in the time integration. We use the \(\mathcal{P }^1\) and \(\mathcal{P }^2\) elements on the meshes with \(64 \times 64\) cells. Figure 14 shows the evolution of the concentration field. The concentration evolution can basically be categorized in two phases, the phase separation (before \(t=8 \times 10^{-6}\)) stage ( \(\varDelta t=2 \times 10 ^{-7}\)) and the coarsening process (from \(t=8 \times 10^{-6}\) onwards) stage ( \(\varDelta t=0.01 \varDelta x^2\)). We can see that these two methods show the same phenomenon and the numerical results show similar patterns with the numerical calculations in Gomez et al. [20].
Fig. 14

The contours of \(u({\mathbf{x}},t)\) for equation (5.12) at different time with \(\mathcal{P }^2\) elements on the uniform mesh with \(64 \times 64\) cells

5.3 Three Space Dimension

Example 8

Convex splitting scheme with degenerate mobility

We consider the scheme (2.4) in \(\varOmega =(0,2\pi ) \times (0,2\pi ) \times (0,2\pi )\) with \(b(u)=1-u^2, \gamma =1, \varPsi (u)=\frac{1}{4}(1-u^2)^2\) and periodic boundary conditions. We take the exact solution of
$$\begin{aligned} u(x,y,z,t)=e^{-3t}\sin (x)\sin (y)\sin (z), \end{aligned}$$
(5.13)
with the source term \(f\), where \(f\) is a given function so that (5.13) is the exact solution.
The convex splitting scheme (2.4) is the first order accuracy in time, so we just consider \(\mathcal{P }^0\) approximation. The \(L^2\) and \(L^\infty \) errors, and the numerical order of accuracy at time \(t=1.0\) with different time steps are presented in Table 10. The numerical experiments go well with the theoretical result of the unconditional energy stability for the scheme.
Table 10

Accuracy test for the scheme (2.4) with the exact solution (5.13)

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

To illustrate the superiority of the FAS MG solver, we present the convergence rates of the method for a single time-step by using \(\mathcal{P }^1\) and \(\mathcal{P }^2\) elements. From Fig. 15, we can see that each iteration of the FAS MG solver is an \(O(N)\) operation. From Table 11 we see that the number of iterations is independent of the mesh size and the Gauss–Seidel smoother has better convergence behavior than the Jacobi smoother. The convergence rates for \(\mathcal{P }^2\) approximation is shown in Fig. 16 and each iteration of the FAS MG solver is also an \(O(N)\) operation.
Fig. 15

Convergence rates of FAS MG solver for \(\mathcal{P }^1\) approximation in three space dimension

Fig. 16

Convergence rates of FAS MG solver for \(\mathcal{P }^2\) approximation with Gauss-Seidel smoother in three space dimension

Table 11

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

We consider the CH equation
$$\begin{aligned} u_t=\nabla \cdot (b(u)\nabla (-\gamma \varDelta u+\varPsi ^{\prime }(u))), \end{aligned}$$
(5.14)
with \(b(u)=1, \varPsi (u)=600(u \mathrm ln u+(1-u)\mathrm ln (1-u))+1800u(1-u),\gamma =1\). The initial condition is
$$\begin{aligned} u_0(x)= {\left\{ \begin{array}{ll} 0.71, &{} {\mathbf{x}} \in \varOmega _1,\\ 0.69, &{} {\mathbf{x}} \in \varOmega _2, \end{array}\right. } \end{aligned}$$
(5.15)
where the square domain
$$\begin{aligned} \varOmega&= (-0.5,0.5)\times (-0.5,0.5)\times (-0.5,0.5), \\ \varOmega _1&= (-0.2,0.2)\times (-0.2,0.2)\times (-0.2,0.2), \quad \varOmega _2=\Omega -\varOmega _1. \end{aligned}$$
The boundary conditions are (5.11).
The implicit third order ARK method and the DIRK method are used in the time integration. We use the \(\mathcal{P }^0\) and \(\mathcal{P }^1\) elements on the meshes with \(64 \times 64 \times 64\) cells, respectively. The time step is taken as \(\varDelta t=2 \times 10 ^{-7}\). The contours at \(t=8 \times 10^{-5}\) are shown in Fig. 17. The numerical results show the similar phenomenon as in the two-dimensional case and we can conclude that the MG solver is efficient for the three-dimensional problems.
Fig. 17

The contours of \(u({\mathbf{x}},t)\) for equation (5.14), \(\mathcal{P }^0\) and \(\mathcal{P }^1\) elements on the uniform mesh with \(64 \times 64 \times 64\) cells

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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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China

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