Improved error estimates for a modified exponential Euler method for the semilinear stochastic heat equation with rough initial data
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A class of stochastic Besov spaces B^p L^2(Ω;Ḣ^α(𝒪)), 1≤ p≤∞ and α∈[-2,2], is introduced to characterize the regularity of the noise in the semilinear stochastic heat equation d u -Δ u d t =f(u) d t + d W(t) , under the following conditions for some α∈(0,1]: ∫_0^te^-(t-s)A d W(s) _L^2(Ω;L^2(𝒪))≤ C t^α/2 ∫_0^te^-(t-s)A d W(s) _B^∞ L^2(Ω;Ḣ^α(𝒪))≤ 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 α=1/2). The latter would fail to satisfy the conditions with α=1/2 if the stochastic Besov norm ·_B^∞ L^2(Ω;Ḣ^α(𝒪)) is replaced by the classical Sobolev norm ·_L^2(Ω;Ḣ^α(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 α in both time and space for possibly nonsmooth initial data in L^4(Ω;Ḣ^β(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.
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