arXiv subjects

Zhihui Liu

Publications and source records attributed to Zhihui Liu.

3 recordsLinked to original sources

arXiv2026

Numerical Ergodicity and Optimal Strong Error Estimates for a Class of Novel Tamed Schemes to Superlinear SPDEs

We construct a class of novel tamed schemes for superlinear stochastic partial differential equations (SPDEs), including the stochastic Allen--Cahn equation driven by either multiplicative or additive noise. The schemes preserve the same Lyapunov structure as the original system, and we rigorously establish their longtime unconditional stability. Furthermore, we prove that the corresponding Galerkin-based fully discrete tamed schemes inherit the unique ergodicity of the underlying SPDEs and achieve optimal strong convergence rates in both the multiplicative and additive noise cases.

math.NA↗

arXiv2025

Numerical Ergodicity and Uniform Estimate of Monotone SPDEs Driven by Multiplicative Noise

We analyze the long-time behavior of numerical schemes for a class of monotone stochastic partial differential equations (SPDEs) driven by multiplicative noise. By deriving several time-independent a priori estimates for the numerical solutions, combined with the ergodic theory of Markov processes, we establish the exponential ergodicity of these schemes with a unique invariant measure, respectively. Applying these results to the stochastic Allen--Cahn equation indicates that these schemes always have at least one invariant measure, respectively, and converge strongly to the exact solution with sharp time-independent rates. We also show that these numerical invariant measures are exponentially ergodic and thus give an affirmative answer to a question proposed in (J. Cui, J. Hong, and L. Sun, Stochastic Process. Appl. (2021): 55--93), provided that the interface thickness is not too small.

math.NA↗

arXiv2021

$L^p$-Convergence Rate of Backward Euler Schemes for Monotone SDEs

We give a unified method to derive the strong convergence rate of the backward Euler scheme for monotone SDEs in $L^p(Ω)$-norm, with general $p \ge 4$. The results are applied to the backward Euler scheme of SODEs with polynomial growth coefficients. We also generalize the argument to the Galerkin-based backward Euler scheme of SPDEs with polynomial growth coefficients driven by multiplicative trace-class noise.

math.NA↗