首页|Improved error estimates for a modified exponential Euler method for the semilinear stochastic heat equation with rough initial data
Improved error estimates for a modified exponential Euler method for the semilinear stochastic heat equation with rough initial data
扫码查看
点击上方二维码区域，可以放大扫码查看
原文链接
万方数据
A class of stochastic Besov spaces BpL2(Ω;Hα(O)),1≤p≤∞ and α ∈[-2,2],is introduced to characterize the regularity of the noise in the semilinear stochastic heat equation du-△udt=f(u)dt+dW(t),under the following conditions for some α ∈(0,1]:||∫toe-(t-s)AdW(s)||L2(Ω;L2(O))≤ Ctα/2 and||∫toe-(t-s)AdW(s)||B∞L2(Ω;(H)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∞L2(Ω;(H)α(O))isreplaced by the classical Sobolev norm||·||L2(Ω;(H)α(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 α in both the time and space for possibly nonsmooth initial data in L4(Ω;(H)β(O))withβ＞-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.
万方数据
