Finite Difference Method for Stochastic Cahn-Hilliard Equation Driven by A Fractional Brownian Sheet

TL;DR

This work analyzes a stochastic Cahn–Hilliard equation driven by a fractional Brownian sheet with anisotropic Hurst parameters, establishing regularity of the mild solution and developing a fully discrete numerical scheme that combines spatial finite differences with a tamed exponential Euler time-stepping. The authors prove that the mild solution enjoys finite $\dot{H}^\beta$-norm moments for $0\le\beta<4H_1+H_2-1$, and they derive a strong convergence rate for the fully discrete scheme of $O\big(h^{1-\epsilon}+\tau^{H_1-\tfrac{1}{8}-\tfrac{\epsilon}{2}}\big)$ in $L^\infty$-norm, for any $\epsilon>0$. The analysis addresses challenges from anisotropic noise, non-invertible Neumann Laplacians, and the unbounded diffusion operator in the nonlinear term by leveraging auxiliary perturbation problems and local Lipschitz bounds on the nonlinear map. Numerical experiments in 1D corroborate the theoretical rates and illustrate the method’s effectiveness, including the white-noise case when $H_1=H_2=\tfrac{1}{2}$. The results provide a solid foundation for numerically solving SCHEs with space-time correlated perturbations and suggest avenues for refinement and extension to higher dimensions and weak convergence theory.

Abstract

The stochastic Cahn-Hilliard equation driven by a fractional Brownian sheet provides a more accurate model for correlated space-time random perturbations. This study delves into two key aspects: first, it rigorously examines the regularity of the mild solution to the stochastic Cahn-Hilliard equation, shedding light on the intricate behavior of solutions under such complex perturbations. Second, it introduces a fully discrete numerical scheme designed to solve the equation effectively. This scheme integrates the finite difference method for spatial discretization with the tamed exponential Euler method for temporal discretization. The analysis demonstrates that the proposed scheme achieves a strong convergence rate of $O\big(h^{1-ε}+τ^{H_1-\frac{1}{8}-\fracε{2}}\big)$, where $ε$ is an arbitrarily small positive constant, providing a solid foundation for the numerical treatment of such equations.

Theorems & Definitions (47)

• Lemma 3.1
• Theorem 3.1
• Lemma 3.2
• proof
• Theorem 3.2
• proof
• Remark 3.3
• Lemma 4.1
• proof
• Proposition 4.2