Strong convergence rate of the explicit adaptive time-stepping methods for stochastic diffusion systems with locally Lipschitz coefficients

TL;DR

This paper introduces an adaptive explicit time-stepping method for stochastic diffusion systems with locally Lipschitz coefficients, achieving strong convergence with optimal order and improved efficiency over existing schemes.

Contribution

It develops a novel adaptive scheme that handles polynomial growth in coefficients and proves its strong convergence with optimal order, outperforming traditional methods.

Findings

Achieves strong convergence order 1/2.

Demonstrates improved efficiency in numerical experiments.

Validates theoretical convergence results with various stochastic systems.

Abstract

This paper proposes an adaptive time-stepping mothods for stochastic diffusion systems whose drift and diffusion coefficients are locally Lipschitz continuous and may exhibit polynomial growth. By controlling the growth of both the drift and diffusion coefficients, we give the choice of the state-dependent adaptive timestep and establish strong convergence of the proposed scheme with the optimal order $1/2$. The performance of the adaptive time-stepping scheme is compared with several widely used explicit and implicit schemes, including tamed EM, truncated EM, and backward EM schemes. Numerical experiments on stiff, non-stiff and high-dimensional stochastic diffusion systems verify the improved computational efficiency of the proposed scheme and validate the theoretical results.