The roughness and smoothness of numerical solutions to the stochastic heat equation
The authors study numerical solutions of the stochastic heat equation \(u_t=u_{xx}+\dot W\) on \((0,1)\) with the Dirichlet boundary condition, driven by the space-time white noise \(W\) with an interest in the quadratic variation (in the space variable) and the quartic variation (in the time variable). The numerical solutions are constructed by a one-step \(\theta\) scheme for \(\theta\in[0,1]\), i.e. the forward Euler (\(\theta=0\)), the backward Euler (\(\theta=1\)) and the Crank-Nicholson scheme (\(\theta=1/2\)) are covered here. It is discussed under what conditions upon \(\theta\) and \(c=k/h^2\) (where \(k\) is the length of the the time step and \(h\) of the space step) the quadratic variation (in the space variable) and quartic variation (in the time variable) converge. The purpose of the paper is to study how the numerical approximations reflect the Hölder properties of the true solutions. It is proved that this is so only for the Crank-Nicholson scheme with \(c=1/(\pi-2)\). The paper contains further results on qualitative properties of variations of the numerical approximations.
stochastic partial differential equations
scientific article; zbMATH DE number 6125020