Improved error estimates for a modified exponential Euler method for the semilinear stochastic heat equation with rough initial data

Abstract

A class of stochastic Besov spaces $B^p L^2(\Omega;\dot H^\alpha(\mathcal{O}))$, $1\le p\le\infty$ and $\alpha\in[-2,2]$, is introduced to characterize the regularity of the noise in the semilinear stochastic heat equation \begin{equation*} {\rm d} u -\Delta u {\rm d} t =f(u) {\rm d} t + {\rm d} W(t) , \end{equation*} under the following conditions for some $\alpha\in(0,1]$: $$ \Big\| \int_0^te^{-(t-s)A}{\rm d} W(s) \Big\|_{L^2(\Omega;L^2(\mathcal{O}))} \le C t^{\frac{\alpha}{2}} \quad\mbox{and}\quad \Big\| \int_0^te^{-(t-s)A}{\rm d} W(s) \Big\|_{B^\infty L^2(\Omega;\dot H^\alpha(\mathcal{O}))}\le C. $$ The conditions above are shown to be satisfied by both trace-class noises (with $\alpha=1$) and one-dimensional space-time white noises (with $\alpha=\frac12$). The latter would fail to satisfy the conditions with $\alpha=\frac12$ if the stochastic Besov norm $\|\cdot\|_{B^\infty L^2(\Omega;\dot H^\alpha(\mathcal{O}))}$ is replaced by the classical Sobolev norm $\|\cdot\|_{L^2(\Omega;\dot H^\alpha(\mathcal{O}))}$, and this often causes reduction of the convergence order in the numerical analysis of the semilinear stochastic heat equation. In this article, the convergence of a modified exponential Euler method, with a spectral method for spatial discretization, is proved to have order $\alpha$ in both time and space for possibly nonsmooth initial data in $L^4(\Omega;\dot{H}^{\beta}(\mathcal{O}))$ with $\beta>-1$, by utilizing the real interpolation properties of the stochastic Besov spaces and a class of locally refined stepsizes to resolve the singularity of the solution at $t=0$.