Pathwise Uniform Convergence of Time Discretisation Schemes for SPDEs

arXiv2303.00411MaRDI QIDQ6509000

Mark Veraar, Katharina Klioba

Abstract: In this paper we prove convergence rates for time discretisation schemes for semi-linear stochastic evolution equations with additive or multiplicative Gaussian noise, where the leading operator $A$ is the generator of a strongly continuous semigroup $S$ on a Hilbert space $X$, and the focus is on non-parabolic problems. The main results are optimal bounds for the uniform strong error ${\displaystyle }$mathrm{E}_{k}^{infty} := Big(mathbb{E} sup_{jin {0, ldots, N_k}} |U(t_j) - U^j|^pBig)^{1/p},${\displaystyle }$ where $pin[2,infty)$, $U$ is the mild solution, $U^j$ is obtained from a time discretisation scheme, $k$ is the step size, and $N_k=T/k$. The usual schemes such as splitting/exponential Euler, implicit Euler, and Crank-Nicolson, etc. are included as special cases. Under conditions on the nonlinearity and the noise we show - $mathrm{E}_k^{infty}lesssimklog(T/k)$ (linear equation, additive noise, general $S$); - $mathrm{E}_k^{infty}lesssimsqrtklog(T/k)$ (nonlinear equation, multiplicative noise, contractive $S$); - $mathrm{E}_k^{infty}lesssimklog(T/k)$ (nonlinear wave equation, multiplicative noise). The logarithmic factor can be removed if the splitting scheme is used with a (quasi)-contractive $S$. The obtained bounds coincide with the optimal bounds for SDEs. Most of the existing literature is concerned with bounds for the simpler pointwise strong error ${\displaystyle }$mathrm{E}_k:=�igg(sup_{jin {0,ldots,N_k}}mathbb{E} |U(t_j) - U^{j}|^p�igg)^{1/p}.${\displaystyle }$ Applications to Maxwell equations, Schr"odinger equations, and wave equations are included. For these equations our results improve and reprove several existing results with a unified method.

Mathematics Subject Classification ID

One-parameter semigroups and linear evolution equations (47D06) Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs (65M12) Stochastic partial differential equations (aspects of stochastic analysis) (60H15) Computational methods for stochastic equations (aspects of stochastic analysis) (60H35) Numerical solutions to stochastic differential and integral equations (65C30) Numerical solutions to abstract evolution equations (65J08)

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