Explicit Third-Order Unconditionally Structure-Preserving Schemes for Conservative Allen–Cahn Equations

Abstract

Compared with the well-known classical Allen–Cahn equation, the modified Allen–Cahn equation, which is equipped with a nonlocal Lagrange multiplier or a local-nonlocal Lagrange multiplier, enforces the mass conservation for modeling phase transitions. In this work, a class of up to third-order explicit structure-preserving schemes is proposed for solving these two modified conservative Allen–Cahn equations. Based on second-order finite-difference space discretization, we investigate the newly developed improved stabilized integrating factor Runge–Kutta (isIFRK) schemes for conservative Allen–Cahn equations. We prove that the original stabilized integrating factor Runge–Kutta schemes fail to preserve the mass conservation law when the stabilizing constant $\kappa > 0$ and the initial mass does not equal zero, while isIFRK schemes not only preserve the maximum principle unconditionally, but also conserve the mass to machine accuracy without any restriction on the time-step size. Convergence of the proposed schemes are also presented. At last, a series of numerical experiments validate that each reformulation of the conservative Allen–Cahn equations has it own advantage, and isIFRK schemes can reach the expected high-order accuracy, conserve the mass, and preserve the maximum principle unconditionally.

Appendix: Other Properties of pRK

Consider an initial value problem for a system of ordinary differential equations (ODEs) of type

$$\begin{aligned} \left\{ \begin{aligned}&u_t = f(t, u(t)), \quad \forall t \in (0, T], \\&u(0) = u^0, \end{aligned} \right. \end{aligned}$$

(41)

where $u^0 \in {\mathbb {R}}^N$, $f: {\mathbb {R}} \times {\mathbb {R}}^N \rightarrow {\mathbb {R}}^N$ such that the problem (41) has a unique solution. Assume that $\Vert \cdot \Vert : {\mathbb {R}}^N \rightarrow {\mathbb {R}}$ is a convex functional

$$\begin{aligned} \Vert \alpha u + (1 - \alpha ) v \Vert \le \alpha \Vert u \Vert + (1 - \alpha ) \Vert v\Vert , \quad \forall \alpha \in [0, 1], \text {and~} u, v \in \mathbb {R}^N. \end{aligned}$$

We present definitions of several properties [3] relevant for the numerical stability of time integration methods for (41).

Definition 5.1

A method is strong stability preserving/monotone with respect to the functional $\Vert \cdot \Vert$ if $\Vert u^{n+1} \Vert \le \Vert u^{n}\Vert , \forall n \ge 0$ under the assumption that

$$\begin{aligned} 0 \le \Vert u + \tau f(t, u)\Vert \le \Vert u\Vert , \quad \forall 0 < \tau \le \tau _{FE}. \end{aligned}$$

(42)

Definition 5.2

(Positivity). A method is positive if, whenever $u^0 \ge 0$, it guarantees that $u^{n+1} \ge 0$ under the assumption that

$$\begin{aligned} u+ \tau f(t, u) \ge 0, \quad \forall 0 < \tau \le \tau _{FE}, u \ge 0. \end{aligned}$$

(43)

Definition 5.3

(Range boundedness). A method is range bounded in [m, M] if, whenever $m \le u^0 \le M$, it guarantees that $m \le u^{n+1} \le M$ under the assumption that

$$\begin{aligned} m \le u + \tau f(t, u) \le M, \quad \forall 0 < \tau \le \tau _{FE}, m \le u \le M. \end{aligned}$$

(44)

Definition 5.4

(Contractivity). A method is contractive if $\Vert u^{n+1} - v^{n+1}\Vert \le \Vert u^n - v^n\Vert$ under assumption that

$$\begin{aligned} \Vert u - v + \tau ( f(t, u) - f(t, v)) \Vert \le \Vert u - v\Vert , \quad \forall 0 < \tau \le \tau _{FE}. \end{aligned}$$

(45)

By introducing a stabilizing term $\kappa u$ to the system (41), we have

$$\begin{aligned} u_t = f(t, u(t)) + \kappa u - \kappa u. \end{aligned}$$

(46)

We show that when $\kappa \ge \frac{1}{\tau _{FE}}$, pRK can unconditionally preserve above properties when f(t, u) satisfies assumptions (42) – (45) in Definitions 5.1 – 5.4.

Theorem 5.5

For the system (46) with $\kappa \ge \frac{1}{\tau _{FE}}$, assume the underlying RK Butcher tableau satisfies Assumption 3.1. Then, the solution obtained from pRK with coefficients ${\hat{d}}_i, {\hat{a}}_{i,j}$ (30) and abscissas $c_i$, unconditionally preserves properties in Definitions 5.1 – 5.4 for any $\tau > 0$.

Proof

Consider the preservation of Contractivity 5.4 as an example. Letting $t_{n, j} = t_n + c_j$ and applying pRK to (46) with different $u^n$ and $v^n$ yield

$$\begin{aligned} u_{n, i}&= \frac{1}{\phi _i(c_i \tau \kappa )} [ u^n + \tau \sum _{j = 0}^{i - 1} a_{i,j} \phi _j (c_j \tau \kappa ) ({f}(t_{n, j}, u_{n,j})+\kappa u_{n,j}) ],\quad i = 1, \cdots , s, \end{aligned}$$

(47)

$$\begin{aligned} v_{n, i}&= \frac{1}{\phi _i(c_i \tau \kappa )} [ v^n + \tau \sum _{j = 0}^{i - 1} a_{i,j} \phi _j (c_j \tau \kappa ) ({f}(t_{n, j}, v_{n,j})+\kappa v_{n,j}) ],\quad i = 1, \cdots , s. \end{aligned}$$

(48)

Let $\kappa := \frac{1}{\tau } \ge \frac{1}{\tau _{FE}}$, multiplying (45) with $\kappa$ gives the circle condition [27]

$$\begin{aligned} \Vert \kappa (u - v) + f(t_{n, j}, u) - f(t_{n, j}, v) \Vert \le \kappa \Vert u - v\Vert , \quad \forall \kappa \ge \frac{1}{\tau _{FE}}. \end{aligned}$$

(49)

Subtracting (48) from (47) gives

$$\begin{aligned} \begin{aligned} u_{n, i} - v_{n, i} =&\frac{1}{\phi _i(c_i \tau \kappa )}\big ( u^n - v^n + \tau \sum _{j = 0}^{i-1} a_{i,j} \phi _j (c_j \tau \kappa ) [f(t_{n, j}, u_{n, j}) + \kappa u_{n, j} \\&- (f(t_{n, j}, v_{n, j}) + \kappa v_{n, j}) ] \big ), i = 1, \cdots , s. \end{aligned} \end{aligned}$$

(50)

Taking norm on both sides of (50), and applying conditions in Assumption 3.1 and the circle condition (49) give

$$\begin{aligned} \begin{aligned} \Vert u_{n, i} - v_{n, i}\Vert&\le \frac{1}{\phi _i(c_i \tau \kappa )}[ \Vert u^n - v^n\Vert + \tau \kappa \sum _{j = 0}^{i-1} a_{i,j} \phi _j (c_j \tau \kappa ) \Vert u_{n, j} - v_{n, j} \Vert ],\quad i = 1, \cdots , s. \ \end{aligned} \end{aligned}$$

Assuming $\Vert u_{n, j} - v_{n, j}\Vert \le \Vert u^n - v^n\Vert , j = 1, \cdots , i-1$, by using the third condition in Assumption 3.1, we derive

$$\begin{aligned} \Vert u_{n, i} - v_{n, i}\Vert&\le \frac{1}{\phi _i(c_i \tau \kappa )}[ \Vert u^n - v^n\Vert + \tau \kappa \sum _{j = 0}^{i-1} a_{i,j} \phi _j (c_j \tau \kappa ) \Vert u^n - v^n\Vert ] \\&= \Vert u^n - v^n \Vert ,\quad i \le s. \ \end{aligned}$$

By using mathematical induction, we obtain $\Vert u^{n+1} - v^{n+1}\Vert \le \Vert u^n - v^n\Vert$. The proofs for preservation of other properties in Definitions 5.1– 5.3 can be performed similarly. $\square$