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 ${B^p}{L^2}({\rm{\Omega}};{\dot{H}^\alpha}({\cal O}))$, 1 ⩽ p ⩽ ∞ and α ∈ [−2, 2], is introduced to characterize the regularity of the noise in the semilinear stochastic heat equation

$$du - \Delta udt = f(u)dt + dW(t),$$

under the following conditions for some α ∈ (0,1]

$${\left\| {\int_0^t {{{\rm{e}}^{- (t - s)A}}} dW(s)} \right\|_{{L^2}(\Omega;{L^2}({\cal O}))}} \leqslant C{t^{{\alpha \over 2}}}\,\,\,\,\,{\rm{and}}\,\,\,\,\,{\left\| {\int_0^t {{{\rm{e}}^{- (t - s)A}}} dW(s)} \right\|_{{B^\infty}{L^2}(\Omega;{{\dot H}^\alpha}({\cal O}))}} \leqslant \,C.$$

The conditions above are shown to be satisfied by both trace-class noises (with α =1) and one-dimensional space-time white noises (with $\alpha = {1 \over 2}$). The latter would fail to satisfy the conditions with $\alpha = {1 \over 2}$ if the stochastic Besov norm ${\left\| {\, \cdot \,} \right\|_{{B^\infty}{L^2}(\Omega;{{\dot H}^\alpha}({\cal O}))}}$ is replaced by the classical Sobolev norm ${\left\| {\, \cdot \,} \right\|_{{L^2}(\Omega;{{\dot H}^\alpha}({\cal O}))}}$, and this often causes reduction of the convergence order in the numerical analysis of the semilinear stochastic heat equation. In this paper, the convergence of a modified exponential Euler method, with a spectral method for spatial discretization, is proved to have order αin both the time and space for possibly nonsmooth initial data in ${L^4}(\Omega;{{\dot H}^\beta}({\cal O}))$ with β > −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.