Higher order approximation of nonlinear SPDEs with additive space-time white noise

TL;DR

This paper proves that approximation schemes based on samples from the stochastic convolution achieve near first-order temporal convergence for nonlinear 1+1D SPDEs with additive white noise, outperforming traditional methods.

Contribution

It demonstrates that sampling from the stochastic convolution yields significantly higher convergence rates for nonlinear SPDEs compared to Wiener increment-based schemes.

Findings

Temporal convergence rate is nearly 1 for schemes based on stochastic convolution.

Spatial convergence rate remains nearly 1/2, consistent with existing literature.

Proves these results for a broad class of nonlinearities, including superlinear growth.

Abstract

We consider strong approximations of $1+1$-dimensional stochastic PDEs driven by additive space-time white noise. It has been long proposed (Davie-Gaines '01, Jentzen-Kloeden '08), as well as observed in simulations, that approximation schemes based on samples from the stochastic convolution, rather than from increments of the underlying Wiener processes, should achieve significantly higher convergence rates with respect to the temporal timestep. The present paper proves this. For a large class of nonlinearities, with possibly superlinear growth, a temporal rate of (almost) $1$ is proven, a major improvement on the rate $1/4$ that is known to be optimal for schemes based on Wiener increments. The spatial rate remains (almost) $1/2$ as it is standard in the literature.