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

TL;DR

This paper analyzes the regularity of solutions and proposes a finite difference numerical scheme with proven convergence for the stochastic Cahn-Hilliard equation driven by a fractional Brownian sheet, enhancing modeling of correlated noise.

Contribution

It introduces a fully discrete finite difference scheme combined with a tamed exponential Euler method and establishes its strong convergence rate for this complex stochastic PDE.

Findings

Proved regularity properties of the mild solution.

Developed a numerical scheme with proven convergence rate.

Achieved effective numerical approximation for the stochastic Cahn-Hilliard equation.

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 $O\big(h^{1-\epsilon}+\tau^{H_1-\frac{1}{8}-\frac{\epsilon}{2}}\big)$, where $\epsilon$ is an arbitrarily small positive constant, providing a solid foundation for the numerical treatment of such equations.