On Efficient Second Order Stabilized Semi-implicit Schemes for the Cahn–Hilliard Phase-Field Equation

Abstract

Efficient and energy stable high order time marching schemes are very important but not easy to construct for the study of nonlinear phase dynamics. In this paper, we propose and study two linearly stabilized second order semi-implicit schemes for the Cahn–Hilliard phase-field equation. One uses backward differentiation formula and the other uses Crank–Nicolson method to discretize linear terms. In both schemes, the nonlinear bulk forces are treated explicitly with two second-order stabilization terms. This treatment leads to linear elliptic systems with constant coefficients, for which lots of robust and efficient solvers are available. The discrete energy dissipation properties are proved for both schemes. Rigorous error analysis is carried out to show that, when the time step-size is small enough, second order accuracy in time is obtained with a prefactor controlled by a fixed power of $1/\varepsilon$, where $\varepsilon$ is the characteristic interface thickness. Numerical results are presented to verify the accuracy and efficiency of proposed schemes.

Appendix: Proof of Lemma 1

Proof

We first write down some inequalities that will be frequently used. The first one is the Holder’s inequality

$$\begin{aligned} \Vert u v w\Vert _{L^s} \le \Vert u\Vert _{L^p} \Vert v\Vert _{L^q} \Vert w\Vert _{L^r}, \quad \forall \ p,q,r\in (0,\infty ],\ \frac{1}{s} = \frac{1}{p} + \frac{1}{q} + \frac{1}{r}. \end{aligned}$$

(A.1)

The second one is the Sobolev inequality

$$\begin{aligned} \Vert u \Vert _{L^q} \le C_s \Vert u \Vert _1, \end{aligned}$$

(A.2)

where $q \in [2, \infty )$ for $d=2; q\in [2, \frac{2d}{d-2}]$ for $d>2; C_s$ is a general constant independent of $\phi$. We can further use Poincare’s inequality to get

$$\begin{aligned} \Vert v \Vert _{L^q} \le C_s \Vert \nabla v \Vert ,\quad \forall v\in L^2_0({\varOmega }). \end{aligned}$$

(A.3)

For $v \in L^2_0({\varOmega })$, we also have following inequality

$$\begin{aligned} \Vert v\Vert ^2 = (\nabla v,\nabla (-{\varDelta })^{-1} v) \le \frac{1}{2\delta } \Vert \nabla v\Vert ^2 + \frac{\delta }{2} \Vert v \Vert _{-1}^2, \end{aligned}$$

(A.4)

where $\delta >0$ is an arbitrary constant.

Now, we begin the proof.

• (i) When $\gamma =1$, we have Cahn–Hilliard equation

  $$\begin{aligned} \phi _{t}+ \varepsilon {\varDelta }^2 \phi =\dfrac{1}{\varepsilon } {\varDelta }f(\phi ). \end{aligned}$$

  (A.5)

  Multiplying (A.5) by $-{\varDelta }^{-1} \phi _t$ and using integration by parts, we get

  $$\begin{aligned} \Vert \phi _t\Vert _{-1}^2 + \frac{\varepsilon }{2}\frac{d}{ d t} \Vert \nabla \phi \Vert ^2 =-\frac{1}{\varepsilon }(f(\phi ),\phi _t) =-\frac{1}{\varepsilon }\frac{d}{d t}\int _{{\varOmega }} F(\phi ) dx. \end{aligned}$$

  (A.6)

  After integrating over [0, T], we obtain

  $$\begin{aligned} \int _{0}^{T}\Vert \phi _t\Vert _{-1}^{2}\mathrm{d}t + E_{\varepsilon }(\phi (T)) = E_{\varepsilon }(\phi ^0) \end{aligned}$$

  (A.7)

  Taking maximum values of terms on the left hand side for $T \in [0, \infty ]$, we get the first part of (i) from (3.7). From the definition of $E_\varepsilon (\phi )$, and assumption (3.1) we know

  $$\begin{aligned} \Vert \phi \Vert _{L^2}^2 \le B_0 |{\varOmega }| + B_1 \varepsilon ^{-\sigma _1+1} \lesssim \varepsilon ^{-(\sigma _1-1)^+}. \end{aligned}$$

  (A.8)

  Combining above estimate with the fact $\frac{\varepsilon }{2}\Vert \nabla \phi \Vert ^2 \lesssim \varepsilon ^{-\sigma _1}$, we get

  $$\begin{aligned} \Vert \phi \Vert _1^2 \lesssim \varepsilon ^{-(\sigma _1+1)}. \end{aligned}$$

  (A.9)

• (ii) We formally differentiate (A.5) in time to obtain

  $$\begin{aligned} \phi _{tt}+ \varepsilon {\varDelta }^2 \phi _{t} = \dfrac{1}{ \varepsilon }{\varDelta }\left( f'(\phi )\phi _t \right) . \end{aligned}$$

  (A.10)

  Pairing (A.10) with $-{\varDelta }^{-1}\phi _t$ and using (A.4), yields

  $$\begin{aligned} \begin{aligned} \frac{1}{2}\frac{d}{d t}\Vert \phi _{t}\Vert _{-1}^2 + \varepsilon \Vert \nabla \phi _{t}\Vert ^2 =&-\dfrac{1}{\varepsilon }\left( f'(\phi )\phi _t, \phi _t \right) \le \frac{\tilde{c}_0}{\varepsilon }\Vert \phi _t\Vert ^2\\ \le \,&\frac{\varepsilon }{2} \Vert \nabla \phi _{t}\Vert ^2 + \frac{\tilde{c}_{0}^2}{2 \varepsilon ^3}\Vert \phi _{t}\Vert _{-1}^2.\\ \end{aligned} \end{aligned}$$

  (A.11)

  Integrating (A.11) over [0, T] and taking maximum values for terms depending on T, we get

  $$\begin{aligned} \mathop {{{\mathrm{ess~sup}}}}\limits _{t\in [0,\infty ]} \Vert \phi _{t}\Vert _{-1}^2+ \varepsilon \int _{0}^{\infty } \Vert \nabla \phi _{t}\Vert ^2\mathrm{d} t \lesssim \frac{\tilde{c}_{0}^2}{ \varepsilon ^3} \int _{0}^{\infty } \Vert \phi _t\Vert _{-1}^2 \mathrm{d} t +\Vert \phi _{t}^0\Vert _{-1}^2. \end{aligned}$$

  (A.12)

  The assertion then follows from (i) and the inequality (3.8) of Assumption 2.

• (iii) Testing (A.10) with $\phi _t$, using (A.1) and (A.2) with Poincare’s inequality, we get

  $$\begin{aligned} \begin{aligned} \frac{1}{2} \frac{d}{d t}\Vert \phi _t\Vert ^2 + \varepsilon \Vert {\varDelta }\phi _t \Vert ^2&= \frac{1}{\varepsilon }(f'(\phi ) \phi _t, {\varDelta }\phi _t) \le \frac{1}{\varepsilon } \Vert f'(\phi )\Vert _{L^3} \Vert \phi _t\Vert _{L^6} \Vert {\varDelta }\phi _t\Vert \\&\le \frac{\varepsilon }{2} \Vert {\varDelta }\phi _t\Vert ^2 +\frac{1}{2 \varepsilon ^3} \Vert f'(\phi )\Vert _{L^3}^2 \Vert \phi _t \Vert _{L^6}^2\\&\le \frac{\varepsilon }{2} \Vert {\varDelta }\phi _t\Vert ^2 +\frac{C_s^2}{2 \varepsilon ^3} \Vert f'(\phi )\Vert _{L^3}^2 \Vert \nabla \phi _t \Vert ^2, \end{aligned} \end{aligned}$$

  (A.13)

  which leads to

  $$\begin{aligned} \mathop {{{\mathrm{ess~sup}}}}\limits _{t\in [0,\infty ]}\Vert \phi _t\Vert ^2 + \varepsilon \int _{0}^{\infty } \Vert {\varDelta }\phi _t\Vert ^2 \mathrm{d} t \lesssim \frac{C_s}{\varepsilon ^3} \mathop {{{\mathrm{ess~sup}}}}_{t\in [0,\infty ]} \Vert f'(\phi )\Vert _{L^3}^2 \int _{0}^{\infty } \Vert \nabla \phi _t\Vert ^2 \mathrm{d} t +\Vert \phi _t^0\Vert ^2.\nonumber \\ \end{aligned}$$

  (A.14)

  On the other hand side, by assumption (3.3), the Sobolev inequality (A.2) and estimate (A.9), we have

  $$\begin{aligned} \Vert f'(\phi )\Vert _{L^3}^2 \lesssim \tilde{c}_2 \Vert \phi \Vert _{L^{3(p-2)}}^{2(p-2)} +\tilde{c}_3 \lesssim \tilde{c}_2 \Vert \phi \Vert _1^{2(p-2)} + \tilde{c}_3 \lesssim \varepsilon ^{-(\sigma _1+1)(p-2)} \end{aligned}$$

  (A.15)

  The assertion then follows from (A.14), (A.15), (ii) and assumption (3.9).

• (iv) Testing (A.10) with $-{\varDelta }^{-1}\phi _{tt}$, we get

  $$\begin{aligned} \Vert \phi _{tt}\Vert _{-1}^2+ \frac{\varepsilon }{2}\frac{d}{dt} \Vert \nabla \phi _{t}\Vert ^2= & {} - \dfrac{1}{\varepsilon }(f'(\phi )\phi _t, \phi _{tt})\nonumber \\= & {} -\frac{1}{2\varepsilon }\frac{d}{dt}(f'(\phi )\phi _t,\phi _t) + \frac{1}{2\varepsilon } (f''(\phi )\phi _t^2,\phi _t) \nonumber \\\le & {} -\frac{1}{2\varepsilon }\frac{d}{dt}(f'(\phi )\phi _t,\phi _t) + \frac{1}{2\varepsilon } \Vert f''\Vert _{L^{6}} \Vert \phi _t^2\Vert _{L^3} \Vert \phi _t\Vert \nonumber \\\le & {} -\frac{1}{2\varepsilon }\frac{d}{dt}(f'(\phi )\phi _t,\phi _t) + \frac{C_s^2}{2\varepsilon } \Vert f''\Vert _{L^{6}} \Vert \nabla \phi _t\Vert ^2 \Vert \phi _t\Vert \nonumber \\ \end{aligned}$$

  (A.16)

  Integrate (A.16) over [0, T], we continue the estimate as

  $$\begin{aligned}&2\int _{0}^{T} \Vert \phi _{tt}\Vert _{-1}^2 \mathrm{d} t + \varepsilon \Vert \nabla \phi _{t}(T)\Vert ^2 - \varepsilon \Vert \nabla \phi _{t}^0\Vert ^2\nonumber \\&\quad \le -\frac{1}{\varepsilon }(f'(\phi )\phi _t,\phi _t)|_{t=T} + \frac{1}{\varepsilon } (f'(\phi ^0)\phi _t^0,\phi _t^0) + \frac{C_s^2}{\varepsilon } \mathop {{{\mathrm{ess~sup}}}}_{t\in [0,T]}\{\Vert f''\Vert _{L^{6}} \Vert \phi _t\Vert \}\int _0^T \Vert \nabla \phi _t\Vert ^2 \mathrm{d}t\nonumber \\&\quad \le \frac{\varepsilon }{2} \Vert \nabla \phi _{t}(T)\Vert ^2 + \frac{\tilde{c}_{0}^2}{2 \varepsilon ^3}\Vert \phi _{t}(T)\Vert _{-1}^2 + \frac{1}{\varepsilon } (f'(\phi ^0)\phi _t^0,\phi _t^0)\nonumber \\&\qquad +\, \frac{C_s^2}{\varepsilon } \mathop {{{\mathrm{ess~sup}}}}_{t\in [0,T]}\{\Vert f''\Vert _{L^{6}} \Vert \phi _t\Vert \}\int _0^T \Vert \nabla \phi _t\Vert ^2\mathrm{d}t, \end{aligned}$$

  (A.17)

  i.e.

  $$\begin{aligned}&2\int _{0}^{T} \Vert \phi _{tt}\Vert _{-1}^2 \mathrm{d} t + \frac{\varepsilon }{2}\Vert \nabla \phi _{t}(T)\Vert ^2 \le \varepsilon \Vert \nabla \phi _{t}^0\Vert ^2 + \frac{1}{\varepsilon } (f'(\phi ^0)\phi _t^0,\phi _t^0)\nonumber \\&\quad +\, \frac{\tilde{c}_{0}^2}{2 \varepsilon ^3}\Vert \phi _{t}(T)\Vert _{-1}^2 + \frac{C_s^2}{\varepsilon } \mathop {{{\mathrm{ess~sup}}}}_{t\in [0,T]} \{\Vert f''\Vert _{L^{6}} \Vert \phi _t\Vert \}\int _0^T \Vert \nabla \phi _t\Vert ^2\mathrm{d}t. \end{aligned}$$

  (A.18)

  On the other hand, by (3.4), the Sobolev inequality (A.2) and estimate (A.9), we have

  $$\begin{aligned} \Vert f''\Vert _{L^{6}} \lesssim \tilde{c}_4\Vert \phi \Vert _{L^{6(p-3)^+}}^{(p-3)^+} + \tilde{c}_5 \lesssim \Vert \phi \Vert _1^{(p-3)^+} \lesssim \varepsilon ^{-\frac{1}{2}(\sigma _1+1)(p-3)^+} \end{aligned}$$

  (A.19)

  By taking maximum for terms depending on T in (A.18) and using (A.19), (ii), (iii) and the inequality (3.10) of Assumption 2. we obtain the assertion (iv).

• (v) We formally differentiate (A.10) in time to derive

  $$\begin{aligned} \phi _{ttt}+ \varepsilon {\varDelta }^2 \phi _{tt} =\dfrac{1}{\varepsilon }{\varDelta }\left( f''(\phi )(\phi _t)^2+f'(\phi )\phi _{tt} \right) . \end{aligned}$$

  (A.20)

  Testing (A.20) with ${\varDelta }^{-2} \phi _{tt}$, we obtian

  $$\begin{aligned}&\frac{1}{2} \frac{d}{ dt } \Vert {\varDelta }^{-1}\phi _{tt}\Vert ^2 + \varepsilon \Vert \phi _{tt} \Vert ^2 =\dfrac{1}{\varepsilon } \left( f''(\phi )(\phi _t)^2+f'(\phi )\phi _{tt}, {\varDelta }^{-1}\phi _{tt}\right) \nonumber \\&\quad \le \dfrac{\varepsilon }{2} \Vert f''(\phi )\Vert ^2_{L^2} \Vert \phi _t\Vert _{L^6}^{4} + \dfrac{1}{2\varepsilon ^3} \Vert {\varDelta }^{-1} \phi _{tt}\Vert ^2_{L^6} +\frac{1}{2\varepsilon ^3}\Vert f'(\phi )\Vert _{L^{3}}^2\Vert {\varDelta }^{-1} \phi _{tt}\Vert _{L^6}^2 +\frac{\varepsilon }{2} \Vert \phi _{tt}\Vert ^2 \nonumber \\&\quad \le \dfrac{\varepsilon }{2} C_s^4\Vert f''(\phi )\Vert ^2_{L^2} \Vert \nabla \phi _t\Vert ^{4} + \dfrac{C_s^2}{2\varepsilon ^3}\Vert \phi _{tt}\Vert ^2_{-1} +\frac{C_s^2}{2\varepsilon ^3} \Vert f'(\phi )\Vert _{L^{3}}^2 \Vert \phi _{tt}\Vert _{-1}^2 +\frac{\varepsilon }{2} \Vert \phi _{tt}\Vert ^2.\nonumber \\ \end{aligned}$$

  (A.21)

  After taking integration from [0, T] and taking maximum for terms depending on T, we have

  $$\begin{aligned}&\mathop {{{\mathrm{ess~sup}}}}\limits _{t\in [0,\infty ]} \Vert {\varDelta }^{-1}\phi _{tt}\Vert ^2 + \varepsilon \int _{0}^{\infty } \Vert \phi _{tt} \Vert ^2 \mathrm{d} t\nonumber \\&\quad \lesssim {} {\varepsilon } \mathop {{{\mathrm{ess~sup}}}}\limits _{t\in [0,\infty ]} \left( \Vert f''(\phi )\Vert ^2_{L^2} \Vert \nabla \phi _t\Vert ^{2}\right) \int _{0}^{\infty } \Vert \nabla \phi _t\Vert ^{2}\mathrm{d} t \nonumber \\&\qquad +\, \frac{1}{\varepsilon ^3} \left( \mathop {{{\mathrm{ess~sup}}}}_{t\in [0,\infty ]} \Vert f'(\phi )\Vert _{L^{3}}^2+1\right) \int _{0}^{\infty } \Vert \phi _{tt}\Vert ^2_{-1}\mathrm{d} t + \Vert {\varDelta }^{-1}\phi _{tt}^0\Vert ^2. \end{aligned}$$

  (A.22)

  The assertion then follows from (A.15), the following estimate

  $$\begin{aligned} \Vert f''\Vert _{L^{2}}^2 \lesssim \tilde{c}_4\Vert \phi \Vert _{L^{2(p-3)^+}}^{(p-3)^+} + \tilde{c}_5 \lesssim \Vert \phi \Vert _1^{2(p-3)^+} \lesssim \varepsilon ^{-(\sigma _1+1)(p-3)^+}, \end{aligned}$$

  (A.23)

  (ii), (iv) and the inequality (3.11) of Assumption 2.

• (vi) Pairing (A.20) with $-{\varDelta }^{-3} \phi _{ttt}$, we obtain

  $$\begin{aligned}&\Vert {\varDelta }^{-1}\phi _{ttt}\Vert _{-1}^2 + \frac{\varepsilon }{2}\frac{d}{d t}\Vert \phi _{tt}\Vert _{-1}^2\nonumber \\&\quad =-\dfrac{1}{\varepsilon }\left( f''(\phi )(\phi _t)^2+f'(\phi )\phi _{tt}, {\varDelta }^{-2} \phi _{ttt} \right) \nonumber \\&\quad \le \dfrac{C_s^2}{\varepsilon ^2} \left( \Vert f''(\phi )\Vert _{L^2}^2 \Vert \phi _{t}\Vert _{L^6}^4 + \Vert f'(\phi )\Vert _{L^3}^2 \Vert \phi _{tt}\Vert ^2 \right) +\frac{1}{2C_s^2}\Vert {\varDelta }^{-2}\phi _{ttt}\Vert _{L^6}^2\nonumber \\&\quad \le \dfrac{C_s^2}{\varepsilon ^2} \left( C_s^4 \Vert f''(\phi )\Vert _{L^2}^2 \Vert \nabla \phi _{t}\Vert ^4 +\Vert f'(\phi )\Vert _{L^3}^2 \Vert \phi _{tt}\Vert ^2 \right) +\frac{1}{2}\Vert {\varDelta }^{-1}\phi _{ttt}\Vert _{-1}^2.\nonumber \\ \end{aligned}$$

  (A.24)

  Integrating (A.24) from $[0,\infty )$, we have

  $$\begin{aligned}&\int _{0}^{\infty } \Vert {\varDelta }^{-1}\phi _{ttt}\Vert _{-1}^2 \mathrm{d} t + \mathop {{{\mathrm{ess~sup}}}}\limits _{t\in [0,\infty ]} \varepsilon \Vert \phi _{tt}\Vert _{-1}^2\nonumber \\&\quad \le \dfrac{2}{\varepsilon ^2} C_s^6 \mathop {{{\mathrm{ess~sup}}}}\limits _{t\in [0,\infty ]} \left( \Vert f''(\phi )\Vert ^2_{L^2}\Vert \nabla \phi _t\Vert ^{2} \right) \int _{0}^{\infty } \Vert \nabla \phi _t\Vert ^{2} \mathrm{d} t\nonumber \\&\qquad +\, \dfrac{2C_s^2}{\varepsilon ^2} \mathop {{{\mathrm{ess~sup}}}}\limits _{t\in [0,\infty ]} \Vert f'(\phi )\Vert _{L^3}^2 \int _{0}^{\infty } \Vert \phi _{tt}\Vert ^2 \mathrm{d} t +\varepsilon \Vert \phi _{tt}^0\Vert _{-1}^2. \end{aligned}$$

  (A.25)

  The assertion then follows from (A.23), (A.15), (ii), (iv), (v) and the inequality (3.12) of Assumption 2.

$\square$