Arbitrarily High-Order Maximum Bound Preserving Schemes with Cut-off Postprocessing for Allen–Cahn Equations

Abstract

We develop and analyze a class of maximum bound preserving schemes for approximately solving Allen–Cahn equations. We apply a kth-order single-step scheme in time (where the nonlinear term is linearized by multi-step extrapolation), and a lumped mass finite element method in space with piecewise rth-order polynomials and Gauss–Lobatto quadrature. At each time level, a cut-off post-processing is proposed to eliminate extra values violating the maximum bound principle at the finite element nodal points. As a result, the numerical solution satisfies the maximum bound principle (at all nodal points), and the optimal error bound $O(\tau ^k+h^{r+1})$ is theoretically proved for a certain class of schemes. These time stepping schemes include algebraically stable collocation-type methods, which could be arbitrarily high-order in both space and time. Moreover, combining the cut-off strategy with the scalar auxiliary value (SAV) technique, we develop a class of energy-stable and maximum bound preserving schemes, which is arbitrarily high-order in time. Numerical results are provided to illustrate the accuracy of the proposed method.

Appendix: Extension to Multi-dimensional Domains

Now we describe the numerical methods in a multi-dimensional rectangular domain $\Omega = (a, b)^d \subset \mathbb {R}^d$, with $d\ge 2$.

Using the setting in one dimension, we denote by $a=x_{0}<x_{1}<\dots <x_{M r}=b$ a partition of the interval [a, b] with a uniform mesh size $\displaystyle h=x_{ir}-x_{(i-1)r} = (b-a)/M$ for all $i=1,\dots ,M$. We let $x_{(i-1)r+j}$ and $w_{j}$, $j=0,\dots ,r$, be the quadrature points and weights of the $(r+1)$-point Gauss–Lobatto quadrature on the subinterval $[x_{(i-1)r}, x_{ir}]$. Moreover, we define the global quadrature weights $w_i$, with $i=0,\ldots , Mr$, by (2.4). Now, $\Omega$ is separated into $M^d$ subrectangles by all grid points $(x_{j_1 r}, \ldots , x_{j_dr})$, with $0 \le j_i \le M$ and $i=1,\dots , d$. We denote this partition by ${\mathcal {T}}_h$, and note that $\displaystyle h$ is the mesh size of the partition ${\mathcal {T}}_h$. Then we apply the tensor-product Lagrange finite elements on the partition ${\mathcal {T}}_h$.

Let $Q^r$ be space of polynomials in the variables ${\fancyscript{x}}_1, \ldots , {\fancyscript{x}}_d$, with real coefficients and of degree at most r in each variable, i.e.,

$$\begin{aligned} Q^r = \Big \{ \sum _{0\le \beta _1,\beta _2,\ldots ,\beta _d\le r} c_{\beta _1 \beta _2\ldots \beta _d} {\fancyscript{x}}_1^{\beta _1} \cdots {\fancyscript{x}}_d^{\beta _d}, \quad \text {with}~~ c_{\beta _1 \beta _2\ldots \beta _d} \in \mathbb {R} \Big \}. \end{aligned}$$

The $H^1$-conforming tensor-product finite element space, denoted by $S_h^r$, is

$$\begin{aligned} S_h^r = \{ v\in H^1(\Omega ): v\vert _K \in Q^r \,\, \text {for all} \,\, K\in {\mathcal {T}}_h \}. \end{aligned}$$

We apply the Gauss–Lobatto quadrature in each subrectangle to approximate of the inner product, i.e.,

$$\begin{aligned} (f,g)_h := \sum _{j_1=0}^{M_1 r}\cdots \sum _{j_d=0}^{M_d r} w_{j_1}\cdots w_{j_d} f(x_{ j_1}, \ldots , x_{j_d}) g(x_{ j_1}, \ldots , x_{j_d}) . \end{aligned}$$

This discrete inner product induces a norm

$$\begin{aligned} \Vert f_h\Vert _h=\sqrt{(f_h,f_h)_h} \quad \forall \, f_h\in S_h^r. \end{aligned}$$

Similarly as the one-dimensional case, the discrete Laplacian $-\Delta _h: S_h^r\rightarrow S_h^r$ is defined such that

$$\begin{aligned} (-\Delta _h v_h, w_h)_h = (\nabla v_h, \nabla w_h)\qquad \text {for all} ~~v_h,w_h\in S_h^r. \end{aligned}$$

(6.1)

Then at n-th time level, with given $u_h^{n-k}, \ldots , u_h^{n-1} \in S_h^r$, we find an intermediate solution ${\hat{u}}_h^n\in S_h^r$ such that

$$\begin{aligned} {\hat{u}}_h^n = \sigma (-\tau \Delta _h) u_h^{n-1} +\tau \sum _{i=1}^m p_i(-\tau \Delta _h) \Big (\sum _{j=1}^{k} L_j (t_{ni}) \Pi _h f(u_h^{n-j})\Big ) \end{aligned}$$

(6.2)

where $t_{ni}=t_{n-1}+c_i\tau$, and $\Pi _h:C({\overline{\Omega }})\rightarrow S_h^r$ is the Lagrange interpolation operator. In order to impose the maximum bound, we apply the cut-off postprocessing: find $u_h^n \in S_h^r$ such that

$$\begin{aligned} u_h^n = \Pi _h \min \big (\max \big ({\hat{u}}_h^n, - \alpha \big ) , \alpha \big ) . \end{aligned}$$

(6.3)

Then the proof of Lemma 3 is still valid in higher dimension. Next, we give the the proof of Lemma 4 which requires some technical argument.

Proof of Lemma 4

To begin with, we let ${\hat{K}} = (0,1)^d$ be a unit reference square and recall that the local Gauss–Lobatto quadrature is accurate for any function $w\in Q^{2r-1}({\hat{K}})$. Using the homogeneous Neumann boundary condition and (6.1), we obtain

$$\begin{aligned} \begin{aligned}&\quad (\Pi _h \Delta v - \Delta _h \Pi _h v, \varphi _h )_h \\&= \Big (( \Delta v , \varphi _h )_h - ( \Delta v , \varphi _h )\Big ) + \Big (( \nabla v , \nabla \varphi _h ) - (\nabla \Pi _h v, \nabla \varphi _h )\Big )\\&=: I + II \end{aligned} \end{aligned}$$

(6.4)

For any $K_{i_1, \dots i_d} = \{{\fancyscript{x}}\in \Omega \subset \mathbb {R}^d : {\fancyscript{x}}_k \in (x_{i_k-1}, x_{i_k}), \text{ for } k = 1,2,\dots ,d\} \in {\mathcal {T}}_h$, there exists a affine transformation from K to $\hat{K}$, which maps $x_i$ to $\hat{x}_i$ ,$v({\fancyscript{x}})$ to $\hat{v}(\hat{{\fancyscript{x}}})$ and $\phi _h({\fancyscript{x}})$ to $\hat{\phi }_h(\hat{{\fancyscript{x}}})$. Then the Bramble–Hilbert lemma leads to

$$\begin{aligned} |I |&= |( \Delta v , \varphi _h )_h - ( \Delta v , \varphi _h ) |\\&= \Big |\sum _{i_1, \dots , i_d=1}^M \Big (Q_{K_{i_1 \dots i_d}}^{GL} (\Delta v \varphi _h) - \int _{K_{i_1 \dots i_d}} (\Delta v ) \varphi _h \,\mathrm {d}x\Big ) \Big |\\&\leqslant Ch^{d-2}\sum _{i_1, \dots , i_d=1}^M \Vert \Delta \hat{v} \hat{\phi }_h\Vert _{W^{2r, 1}(\hat{K})} \leqslant Ch^{d-2}\sum _{i_1, \dots , i_d=1}^M |\Delta \hat{v} \hat{\phi }_h|_{W^{2r, 1}(\hat{K})} \\&\le C h^{2r} \sum _{i_1, \dots , i_d=1}^M |\Delta v \varphi _h |_{W^{2r,1}(K_{i_1\dots i_d})} \\&\le C h^{2r} \sum _{i=1}^M \Vert v \Vert _{H^{2r+2}(K_{i_1, \dots i_d})} \Vert \varphi _h \Vert _{H^{r}(K_{i_1 \dots i_d})} \end{aligned}$$

Then we apply the inverse inequality to derive

$$\begin{aligned} |I |&\le Ch^{r+1} \sum _{i=1}^M \Vert v \Vert _{H^{2r+2}(K_{i_1, \dots i_d})} \Vert \varphi _h \Vert _{H^{1}(K_{i_1 \dots i_d})} \\&\le Ch^{r+1} \Vert v \Vert _{H^{2r+2}(\Omega )} \Vert \varphi _h \Vert _{H^{1}(\Omega )}. \end{aligned}$$

Similar argument also leads to the estimate for the term II in (6.4):

$$\begin{aligned} II&= ( \nabla (v- \Pi _h v), \nabla \varphi _h ) = \sum _{K\in {\mathcal {K}}}^M \int _{K} \nabla (v- \Pi _h v)\cdot \nabla \varphi _h \,\mathrm {d}{\fancyscript{x}}. \end{aligned}$$

The case $r=1$ has been studied in [35, Theorem 4.1], so we only consider the cases that $r\ge 2$. Here we claim that

$$\begin{aligned} \int _K\nabla (w- \Pi _h w)\cdot \nabla \varphi _h \,\mathrm {d}{\fancyscript{x}}= 0\quad \text {for any} ~~v \in P^{r+1}(K), \phi _h \in Q^{r}(K). \end{aligned}$$

To show this, we only need to show it works on the reference square ${\hat{K}}=(0,1)^d$, and affine transformations will generalize the result to any triangle K. Furthermore, by the symmetry, we only need to show that

$$\begin{aligned} B(w, \psi _h) := \int _{{\hat{K}}} \partial _{{\fancyscript{x}}_1} (w- \Pi _h w)\cdot \partial _{{\fancyscript{x}}_1} \psi _h \,\mathrm {d}{\fancyscript{x}}= 0, \end{aligned}$$

(6.5)

for any $w\in P^{r+1}({\hat{K}}), \psi _h \in Q^{r}({\hat{K}})$.

Note that $\Pi _h w \in Q^r({\hat{K}})$. Then for any $w \in P^{r+1}({\hat{K}}) \cap Q^{r}({\hat{K}})$, $w = \Pi w$, so $B(w, \psi _h) = 0$ for all $w\in P^{r+1}({\hat{K}}) \cap Q^{r}({\hat{K}}), \psi _h \in Q^r(e)$. Moreover, for $w({\fancyscript{x}}) = ({\fancyscript{x}}_k)^{r+1}$ with $k \ne 1$, we have $\partial _{{\fancyscript{x}}_1} (w- \Pi _h w) = 0$ which also leads to $B(w, \psi _h) = 0$.

For $w = {\fancyscript{x}}_1^{r+1}$, and $\psi _h = {\fancyscript{x}}_1^{k_1}\cdot \cdots \cdot {\fancyscript{x}}_d^{k_d}$, $k_i\leqslant r$ which is a basis of $Q^{r}({\hat{K}})$,

$$\begin{aligned} B(w, \psi _h)&= \int _0^1 \partial _{{\fancyscript{x}}_1} ({\fancyscript{x}}_1^{r+1}- \Pi _h {\fancyscript{x}}_1^{r+1})\partial _{{\fancyscript{x}}_1}({\fancyscript{x}}_1^{k_1})\mathrm {d}{\fancyscript{x}}_1 \int _0^1{\fancyscript{x}}_2^{k_2}\mathrm {d}{\fancyscript{x}}_2\cdots \int _0^1{\fancyscript{x}}_d^{k_d}\mathrm {d}{\fancyscript{x}}_d\\&= \prod _{i=2}^d\left( \frac{1}{1+k_i}\right) \int _0^1 \partial _{{\fancyscript{x}}_1} ({\fancyscript{x}}_1^{r+1}- \Pi _h {\fancyscript{x}}_1^{r+1})\partial _{{\fancyscript{x}}_1}({\fancyscript{x}}_1^{k_1})\mathrm {d}{\fancyscript{x}}_1\\&= -\prod _{i=2}^d\left( \frac{1}{1+k_i}\right) \int _0^1 ({\fancyscript{x}}_1^{r+1}- \Pi _h {\fancyscript{x}}_1^{r+1})\partial _{{\fancyscript{x}}_1}^2({\fancyscript{x}}_1^{k_1})\mathrm {d}{\fancyscript{x}}_1. \end{aligned}$$

For $k_1 \leqslant 1$, $\partial _{{\fancyscript{x}}_1}^2{\fancyscript{x}}_i^{k_1}=0$, so $B(w, \psi ) = 0$. For $k_1 \geqslant 2$, we can get

$$\begin{aligned} \begin{aligned} \int _0^1 ({\fancyscript{x}}_1^{r+1}- \Pi _h {\fancyscript{x}}_1^{r+1})&\partial _{{\fancyscript{x}}_1}^2({\fancyscript{x}}_1^{k_1})\mathrm {d}{\fancyscript{x}}_1 = k_1(k_1-1)\int _0^1 ({\fancyscript{x}}_1^{r+1}- \Pi _h {\fancyscript{x}}_1^{r+1}){\fancyscript{x}}_1^{k_1-2}\mathrm {d}{\fancyscript{x}}_1\\&= k_1(k_1-1)\int _0^1 ({\fancyscript{x}}_1^{r+k_1-1}- {\fancyscript{x}}_1^{k_1-2}\Pi _h {\fancyscript{x}}_1^{r+1})\mathrm {d}{\fancyscript{x}}_1. \end{aligned} \end{aligned}$$

Note that ${\fancyscript{x}}_1^{r+k_1-1}$ is a polynomial of order at most $2r-1$, ${\fancyscript{x}}_1^{k_1-2}\Pi _h {\fancyscript{x}}_1^{r+1}$ is a polynomial of order at most $2r-2$, and ${\fancyscript{x}}_1^{r+k_1-1} - {\fancyscript{x}}_1^{k_1-2}\Pi _h {\fancyscript{x}}_1^{r+1} = 0$ at quadrature points. We know that

$$\begin{aligned} \int _0^1 ({\fancyscript{x}}_1^{r+k_1-1}- {\fancyscript{x}}_1^{k_1-2}\Pi _h {\fancyscript{x}}_1^{r+1})\mathrm {d}{\fancyscript{x}}_1 = 0. \end{aligned}$$

Therefore $B(w, \psi _h) = 0$ for $w={\fancyscript{x}}_1^{r+1}$, $\psi _h \in Q^r({\hat{K}})$.

As a result, we conclude that $B(w, \psi _h) = 0$ for all $\psi _h\in Q^r({\hat{K}})$, $w \in P^{r+1}({\hat{K}})\cap Q^r({\hat{K}})$ or $w = {\fancyscript{x}}_k^{r+1}$ for $1\leqslant k \leqslant d$, and hence

$$\begin{aligned} B(w, \psi _h) = 0,\quad \forall w \in P^{r+1}({\hat{K}}), \psi _h \in Q^r({\hat{K}}). \end{aligned}$$

Together with Bramble–Hilbert lemma, we derive that for any $K \in {\mathcal {T}}_h$

$$\begin{aligned} |\int _{K} \nabla (v- \Pi _h v)\cdot \nabla \varphi _h \,\mathrm {d}{\fancyscript{x}}|&=h^{d-2} |\int _{\hat{K}} \nabla (\hat{v}- \Pi _h \hat{v})\cdot \nabla \hat{\varphi }_h \,\mathrm {d}{\fancyscript{x}}|\\&\leqslant C h^{d-2} |\hat{v} |_{H^{r+2}(\hat{K})}\Vert \nabla \hat{\phi }_h\Vert _{L^2(\hat{K})} \\&\leqslant C h^{r+1} |v |_{H^{r+2}(K)} \Vert \nabla \phi _h\Vert _{L^2(K)}. \end{aligned}$$

Do the summation over all subrectangle we arrive at

$$\begin{aligned} |II |\leqslant C h^{r+1}\Vert v\Vert _{H^{r+2}(\Omega )}\Vert \phi _h\Vert _{H^1(\Omega )}. \end{aligned}$$

This completes the proof of Lemma 4. $\square$

Then Lemmas 3 and 4 immediately implies Theorem 5 in higher dimensions.