An explicit adaptive time-stepping scheme for superlinear stochastic diffusion systems

Abstract

This paper develops an adaptive time-stepping Euler--Maruyama (EM) scheme for stochastic diffusion systems with superlinearly growing coefficients. The adaptive timestep is chosen according to the superlinear growth of both drift and diffusion coefficients. To prevent excessively small timesteps, a truncated EM scheme is employed as a backstop whenever the adaptive timestep falls below a prescribed threshold. By combining the stochastic analysis with the stopping time technique, we establish the strong convergence of the proposed method and obtain the optimal $1/2$-order strong convergence rate in the $L^q$-sense for $q>2$. {Finally, numerical experiments are carried out for stiff, nonstiff, and stochastic Lorenz systems to validate the theoretical findings. The results indicate that the proposed scheme achieves superior accuracy and performance compared to various fixed-step and adaptive alternatives.