Pathwise Uniform Convergence of Time Discretisation Schemes for SPDEs

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 $${\mathrm{E}}_k^{\infty }:=(\mathbb{E}\underset{j\in \{0,\dots ,N_k\}}{\sup }\Vert U(t_j)-U^j{\Vert }^p{)}^{1/p},$$ where $p\in [2,\infty )$, $U$ 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 the exponential Euler, the implicit Euler, and the Crank-Nicolson method, etc. are included as special cases. Under conditions on the nonlinearity and the noise, we show - ${\mathrm{E}}_k^{\infty }\lesssim k\sqrt{\log (T/k)}$ (linear equation, additive noise, general $S$); - ${\mathrm{E}}_k^{\infty }\lesssim \sqrt{k}\sqrt{\log (T/k)}$ (nonlinear equation, multiplicative noise, contractive $S$); - ${\mathrm{E}}_k^{\infty }\lesssim k\sqrt{\log (T/k)}$ (nonlinear wave equation, multiplicative noise) for a large class of time discretisation schemes. The logarithmic factor can be removed if the exponential Euler method 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 $${\mathrm{E}}_k:=(\underset{j\in \{0,\dots ,N_k\}}{\sup }\mathbb{E}\Vert U(t_j)-U^j{\Vert }^p{)}^{1/p}.$$ Applications to Maxwell equations, Schrödinger equations, and wave equations are included. For these equations, our results improve and reprove several existing results with a unified method and provide the first results known for the implicit Euler and the Crank-Nicolson method.

Accepted for publication in IMA Journal of Numerical Analysis. 52 pages, 1 figure, added Subsection 6.5 with numerical experiments, changed Proposition 2.3, improved all logarithmic to square-root-logarithmic correction factors

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Cited by in corpus (1)

• Strong convergence rates for a full discretization of stochastic wave equation with nonlinear damping