Xueqi Wen | Searcharxiv

arXiv2026

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

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.

math.NA↗

arXiv2026

Adaptive Time-Stepping Euler--Maruyama Scheme for SDEs with Non-Globally Lipschitz Coefficients: Uniform Convergence, Stability and Ergodicity

This paper develops an adaptive time-stepping Euler--Maruyama scheme for stochastic differential equations (SDEs) with non-globally Lipschitz drift and diffusion coefficients. By dynamically adjusting the timestep at each iteration, the proposed scheme effectively prevents numerical instability. We prove the moment boundedness of the numerical solution and establish a $1/2$-order strong convergence rate both on finite-time intervals and uniformly in time. Furthermore, the scheme faithfully inherits the $p$th moment exponential stability of the underlying SDE. For long-time ergodic dynamics, we establish the polynomial ergodicity of the numerical invariant measure. Moreover, we show that the numerical invariant measure converges to the invariant measure of the underlying SDE at an optimal rate of $1/2$ in the $L^q$-Wasserstein distance. Numerical experiments confirm our theoretical results and indicate the superior accuracy and computational performance of the proposed scheme over several fixed-step and existing adaptive methods.

math.NA↗