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.

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

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Cited by in corpus (1)

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