Simple maximum principle preserving time-stepping methods for time-fractional Allen-Cahn equation

Abstract

Two fast L1 time-stepping methods, including the backward Euler and stabilized semi-implicit schemes, are suggested for the time-fractional Allen-Cahn equation with Caputo’s derivative. The time mesh is refined near the initial time to resolve the intrinsically initial singularity of solution, and unequal time steps are always incorporated into our approaches so that a adaptive time-stepping strategy can be used in long-time simulations. It is shown that the proposed schemes using the fast L1 formula preserve the discrete maximum principle. Sharp error estimates reflecting the time regularity of solution are established by applying the discrete fractional Grönwall inequality and global consistency analysis. Numerical experiments are presented to show the effectiveness of our methods and to confirm our analysis.

Appendix. Discrete fractional Grönwall lemma

We recall the recently developed discrete fractional Grönwall inequality in [26], which is applicable for any nonuniform time meshes and suitable for a variety of discrete fractional derivatives having a discrete form of ${\sum }_{k=1}^{n}A_{n-k}^{(n)}\triangledown _{\tau } v^{k}$. The following fractional Grönwall lemma, involving the well-known Mittag–Leffler function $E_{\alpha }(z):={\sum }_{k=0}^{\infty }\frac {z^{k}}{\Gamma (1+k\alpha )}$, gathers three previous (slightly simplified) results from [26, Lemma 2.2, Theorems 3.1 and 3.2].

Lemma A.1

For n = 1, 2,⋯ ,N, assume that the discrete convolution kernels $\{A_{n-k}^{(n)}\}_{k=1}^{n}$ satisfy the following two assumptions:

• Ass1. There is a constant π_a > 0 such that $A_{n-k}^{(n)}\geq \frac {1}{\pi _{a}}{\int \limits }_{t_{k-1}}^{t_{k}}\frac {\omega _{1-\alpha }(t_{n}-s)}{\tau _{k}} s$ for $1\leqslant k\leqslant n$.

• Ass2. The discrete kernels are monotone, i.e., $A_{n-k-1}^{(n)}-A_{n-k}^{(n)}\geq 0$ for $1\leqslant k\leqslant n-1$.

Define also a sequence of discrete complementary convolution kernels $\{p_{n-j}^{(n)}\}_{j=1}^{n}$ by

$$ p_{0}^{(n)}:=\frac{1}{A_{0}^{(n)}},\quad p_{n-j}^{(n)}:= \frac{1}{p_{0}^{(j)}} \sum\limits_{k=j+1}^{n}\left( {A_{k-j-1}^{(k)}-A_{k-j}^{(k)}}\right)p_{n-k}^{(n)}, \quad 1\leqslant j\leqslant n-1. $$

(A.1)

Then the discrete complementary kernels $p^{(n)}_{n-j}\ge 0$ are well-defined and fulfill

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{j=k}^{n}p^{(n)}_{n-j}A_{j-k}^{(j)}= 1, \quad\text{ for } 1\le k\le n\le N. \end{array} $$

(A.2)

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{j=1}^{n}p^{(n)}_{n-j}\omega_{1+m\alpha-\alpha}(t_{j})\leqslant \pi_{a}\omega_{1+m\alpha}(t_{n}), \text{ for } m=0,1 \text{ and } 1\le n\le N. \end{array} $$

(A.3)

Suppose that the offset parameter $0 \leqslant \nu <1$, λ is a nonnegative constant independent of the time steps and the maximum step size $\tau \le 1/\sqrt [\alpha ]{2{\Gamma }(2-\alpha )\lambda \pi _{a}}.$ If the nonnegative sequences $(v^{k})_{k=0}^{N}$, $(\xi ^{k})_{k=1}^{N}$ and $(\eta ^{k})_{k=1}^{N}$ satisfy

$$ {\sum}_{k=1}^{n}A_{n-k}^{(n)}\triangledown_{\tau} v^{k}\le \lambda v^{n-\nu}+\xi^{n}+\eta^{n}\quad\text{ for } 1\le n\le N, $$

(A.4)

then it holds that, for 1 ≤ n ≤ N,

$$ \begin{array}{@{}rcl@{}} v^{n}&\le&2E_{\alpha}\left( 2\max\{1,\rho\}\lambda\pi_{a} t_{n}^{\alpha}\right) \left( v^{0}+\max_{1\le k\le n}{\sum}_{j=1}^{k}p^{(k)}_{k-j}(\xi^{j}+\eta^{j})\right)\\ &\le&2E_{\alpha}\left( 2\max\{1,\rho\}\lambda\pi_{a} t_{n}^{\alpha}\right)\\ &&\times \left( v^{0}+{\Gamma}(1-\alpha)\pi_{a}\max_{1\le k\le n}\{t_{k}^{\alpha}\xi^{k}\}+\pi_{a}\omega_{1+\alpha}(t_{n})\max_{1\le k\le n}\eta^{k}\right). \end{array} $$