Second Order Linear Energy Stable Schemes for Allen-Cahn Equations with Nonlocal Constraints

Abstract

We present a set of linear, second order, unconditionally energy stable schemes for the Allen-Cahn equation with nonlocal constraints that preserves the total volume of each phase in a binary material system. The energy quadratization strategy is employed to derive the energy stable semi-discrete numerical algorithms in time. Solvability conditions are then established for the linear systems resulting from the semi-discrete, linear schemes. The fully discrete schemes are obtained afterwards by applying second order finite difference methods on cell-centered grids in space. The performance of the schemes are assessed against two benchmark numerical examples, in which dynamics obtained using the volume-preserving Allen-Cahn equations with nonlocal constraints is compared with those obtained using the classical Allen-Cahn as well as the Cahn-Hilliard model, respectively, demonstrating slower dynamics when volume constraints are imposed as well as their usefulness as alternatives to the Cahn–Hilliard equation in describing phase evolutionary dynamics for immiscible material systems while preserving the phase volumes. Some performance enhancing, practical implementation methods for the linear energy stable schemes are discussed in the end.

Appendices

Appendix

Sherman-Morrison Formula and its Application to Solving the Integro-Differential Equation

Here we give a brief review on the Sherman-Morrison formula [52] and explain its applications in the practical implementation of our various relevant schemes.

Suppose A is an invertible square matrix, and u,v are column vectors. Then $A+uv^T$ is invertible iff $1+v^TA^{-1}u \ne 0$. If $A+uv^T$ is invertible, then its inverse is given by

$$\begin{aligned} (A+uv^T)^{-1}=A^{-1}-\frac{A^{-1}uv^TA^{-1}}{1+v^TA^{-1}u}. \end{aligned}$$

(A.1)

So if $Ay=b$ and $Az=u$, $(A+uv^T)x=b$ has the solution given by

$$\begin{aligned} x=y-\frac{v^Ty}{1+v^Tz}z. \end{aligned}$$

(A.2)

For the integral term(s) in the semi-discrete schemes in this study such as (4.26), we need to discretize it properly. $\forall f$, we discretize $\int _\Omega f \mathrm {d\mathbf{x}}$ using the composite trapezoidal rule and adding all the elements of the new matrix $w_1 w_2^T f$, where $w_1=\frac{h_x}{2}S$, $w_2=\frac{h_y}{2}S$, $h_x$, $h_y$ are the spatial step sizes and $S={[1,2,2,...,2,2,1]}^T$. For convenience, we use $w_1w_2^T f$ to represent the integral discretized by the composite trapezoidal rule.

To solve Eq. (4.26), we discretize the integral or the inner product of functions $(c,{\upphi }^{n+1})d$ as $u { v}^T {\upphi }^{n+1}$. The scheme is recast to $A{\upphi }^{n+1}+u { v}^T {\upphi }^{n+1}=b^n$. After using the Sherman-Morrison formula, we get

$$\begin{aligned} {\upphi }^{n+1}= A^{-1} {b}^n-\frac{{v}^T{ A}^{-1} { {b}}^n}{{1+{ v}^T A}^{-1} u}{ A}^{-1} u, \end{aligned}$$

(A.3)

In the inner product of vectors, (4.26) can be rewritten into

$$\begin{aligned} {{\upphi }} ^{n+1}= A^{-1}{b}^n-\frac{\langle c,A^{-1}b^n\rangle }{1+\langle c,A^{-1}d\rangle }{A}^{-1}d. \end{aligned}$$

(A.4)

So, indeed the approach we take in the study using discrete inner product is essentially equivalent to applying the Sherman-Morrison formula.

The Energy Dissipation Theorem in the Full Discrete Scheme

Here we only give the energy dissipation theorem in the full discrete scheme for the Allen-Cahn model with a penalizing potential, since the others are similar.

Theorem B.1

The full discrete scheme in (4.23) obeys the following energy dissipation law

$$\begin{aligned} F^{n+1}-F^n=-{\updelta }t\langle {{\tilde{\mu }}}^{n+1/2}, {\overline{M}}^{n+1/2}{{\tilde{\mu }}}^{n+1/2} \rangle . \end{aligned}$$

(B.1)

Hence, it is unconditionally stable.

Proof

Taking the $l^2$ inner product of $\frac{{\upphi }^{n+1}-{\upphi }^n}{{\updelta }t}$ with $-{{\tilde{\mu }}}^{n+1/2}$, we obtain

$$\begin{aligned} \begin{aligned} - \left\langle \frac{{\upphi }^{n+1}-{\upphi }^n}{{\updelta }t},{{\tilde{\mu }}}^{n+1/2} \right\rangle&=\langle \overline{M}^{n+1/2}[{\mu }^{n+1/2}+\sqrt{{\upeta }} {{\upzeta }}^{n+1/2}],{\mu }^{n+1/2}+\sqrt{{\upeta }} {{\upzeta }}^{n+1/2} \rangle \\&= \left\| \sqrt{\overline{M}^{n+1/2}}\left( {\mu }^{n+1/2}+\sqrt{{\upeta }} {{\upzeta }}^{n+1/2}\right) \right\| ^2_d, \end{aligned} \end{aligned}$$

(B.2)

Taking the $l^2$ inner product of ${{\tilde{\mu }}}^{n+1/2}$ with $\frac{{\upphi }^{n+1}-{\upphi }^n}{{\updelta }t}$, we obtain

$$\begin{aligned} \begin{aligned}&\left\langle ({-{\upgamma }_1 \nabla _h ^2{\upphi }+2{\upgamma }_2{\upphi })}^{n+1/2}+2{q}^{n+1/2} \overline{q'}^{n+1/2}+\sqrt{{\upeta }}{{\upzeta }}^{n+1/2},\frac{{\upphi }^{n+1}-{\upphi }^n}{{\updelta }t} \right\rangle \\&\quad =\frac{{\upgamma }_1}{2{\updelta }t}(||\nabla _h {\upphi }^{n+1}||^2_d-||\nabla _h {\upphi }^n||^2_d)+\frac{{\upgamma }_2}{ {\updelta }t}(||{\upphi }^{n+1}||^2_d-||{\upphi }^n||^2_d) \\&\qquad +\left\langle 2{q}^{n+1/2} \overline{q'}^{n+1/2},\frac{{\upphi }^{n+1}-{\upphi }^n}{{\updelta }t} \right\rangle + \left\langle \sqrt{{\upeta }}{{\upzeta }}^{n+1/2},\frac{{\upphi }^{n+1}-{\upphi }^n}{{\updelta }t} \right\rangle . \end{aligned} \end{aligned}$$

(B.3)

Taking the $l^2$ inner product of $q^{n+1}-q^n$ with $\frac{q^{n+1}+q^n}{{\updelta }t}$, we obtain

$$\begin{aligned} \frac{1}{{\updelta }t}(||q^{n+1}||^2_d-||q^{n}||^2_d)=\frac{1}{{\updelta }t}\langle \overline{q'}^{n+1/2}({\upphi }^{n+1}-{\upphi }^n),q^{n+1}+q^{n} \rangle . \end{aligned}$$

(B.4)

Taking the $l^2$ inner product of ${\upzeta }^{n+1}-{\upzeta }^n$ with $\frac{{\upzeta }^{n+1}+{\upzeta }^n}{{\updelta }t}$, we obtain

$$\begin{aligned} \frac{1}{{\updelta }t}(||{\upzeta }^{n+1}||^2_d-||{\upzeta }^{n}||^2_d)=\frac{1}{{\updelta }t}\langle \sqrt{{\upeta }} ||{\upphi }^{n+1}-{\upphi }^n||^2_d,{\upzeta }^{n+1}+{\upzeta }^{n} \rangle . \end{aligned}$$

(B.5)

Combining the above equations, we have

$$\begin{aligned} \begin{aligned}&\frac{{\upgamma }_1}{2{\updelta }t}(||\nabla _h {\upphi }^{n+1}||^2_d-||\nabla _h {\upphi }^n||^2_d)+\frac{{\upgamma }_2}{ {\updelta }t}(||{\upphi }^{n+1}||^2_d-||{\upphi }^n||^2_d) \\&\qquad +\frac{1}{{\updelta }t}(||q^{n+1}||^2_d-||q^{n}||^2_d)+\frac{1}{2 {\updelta }t}(||{\upzeta }^{n+1}||^2_d-||{\upzeta }^{n}||^2_d)\\&\quad =- \left\| \sqrt{\overline{M}^{n+1/2}}({\mu }^{n+1/2}+\sqrt{{\upeta }} {{\upzeta }}^{n+1/2}) \right\| ^2_d. \end{aligned} \end{aligned}$$

(B.6)

This proves the energy stability equality. $\square$