It is known from Beccari et al. (2019) that the standard explicit Euler-type scheme (such as the exponential Euler and the linear-implicit Euler schemes) with a uniform timestep, though computationally efficient, may diverge for the stochastic Allen–Cahn equation. To overcome the divergence, this paper proposes and analyzes adaptive time-stepping schemes, which adapt the timestep at each iteration to control numerical solutions from instability. The a priori estimates in $\mathscr{C}(O)$-norm and $\dot{H}^\beta(O)$-norm of numerical solutions are established provided the adaptive timestep function is suitably bounded, which plays a key role in the convergence analysis. We show that the adaptive time-stepping schemes converge strongly with order  $\beta/2$ in time and $\beta/d$ in space with $d (d=1,2,3)$ being the dimension and $\beta\in(0,2]$. Numerical experiments show that the adaptive time-stepping schemes are simple to implement and at a lower computational cost than a scheme with the uniform timestep.

Publication:

IMA Journal of Numerical Analysis, drae009

https://doi.org/10.1093/imanum/drae009

Authors:

Chuchu Chen

LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China

Email: chenchuchu@lsec.cc.ac.cn

Tonghe Dang

LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China

Jialin Hong

LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China

Email: hjl@lsec.cc.ac.cn