Density convergence of a fully discrete finite difference method for stochastic Cahn--Hilliard equation

TL;DR

This paper establishes the density convergence of a fully discrete finite difference scheme for the stochastic Cahn--Hilliard equation driven by multiplicative noise, addressing a key open problem in numerical density estimation.

Contribution

It introduces a novel localization technique and derives the strong convergence rate, enabling the numerical density to converge in $L^1$ to the true density.

Findings

Proves density convergence in $L^1(\

Estimates the total variation distance between exact and numerical solutions.

Partially resolves an open problem on numerical density computation for stochastic PDEs.

Abstract

This paper focuses on investigating the density convergence of a fully discrete finite difference method when applied to numerically solve the stochastic Cahn--Hilliard equation driven by multiplicative space-time white noises. The main difficulty lies in the control of the drift coefficient that is neither globally Lipschitz nor one-sided Lipschitz. To handle this difficulty, we propose a novel localization argument and derive the strong convergence rate of the numerical solution to estimate the total variation distance between the exact and numerical solutions. This along with the existence of the density of the numerical solution finally yields the convergence of density in $L^1(\mathbb{R})$ of the numerical solution. Our results partially answer positively to the open problem emerged in [J. Cui and J. Hong, J. Differential Equations (2020)] on computing the density of the exact solution numerically.