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

A class of stochastic Besov spaces BpL(2)(Omega; H-alpha(O)), 1 <= p <=infinity and alpha is an element of[-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 alpha is an element of(0,1]: parallel to integral(t)(0)e(-(t-s)A)dW((s))parallel to(2)(L)(Omega;L2(O)) <= Ct(alpha 2 )and parallel to integral(e-(t-s)AdW)(0 (R))(s)parallel to(B)infinity L-2(Omega; H alpha(O)) <= 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=12). The latter would fail to satisfy the conditions with alpha=12if the stochastic Besov norm & Vert;<middle dot>& Vert;B infinity L2(Omega; H alpha(O)) is replaced by the classical Sobolev norm & Vert;<middle dot>& Vert;L2(Omega; H alpha(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 alpha in both the time and space for possibly non-smooth initial data in L-4(Omega; H beta(O)) with beta>-1, by utilizing the real interpolation properties of the stochastic Besov spaces and a class of locally refined step-sizes to resolve the singularity of the solution at t=0.