Strong convergence of adaptive time-stepping schemes for the stochastic Allen–Cahn equation

Abstract

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 $\mathscr{C}(\mathscr{O})$-norm and $\dot{H}^{\beta }(\mathscr{O})$-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 $\frac{\beta }{2}$ in time and $\frac{\beta }{d}$ 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.