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

Abstract

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.

To appear in Journal of Computational Mathematics, doi:10.4208/jcm.2409-m2024-0041

References in corpus (4)

• Rate of Convergence of Space Time Approximations for stochastic evolution equations
• Optimal Strong Rates of Convergence for a Space-Time Discretization of the Stochastic Allen-Cahn Equation with multiplicative noise
• $L^p$-Convergence Rate of Backward Euler Schemes for Monotone SDEs
• Numerical Unique Ergodicity of Monotone SDEs driven by Nondegenerate Multiplicative Noise

Cited by in corpus (1)

• Numerical Unique Ergodicity of Monotone SDEs driven by Nondegenerate Multiplicative Noise