Finite difference method for stochastic Cahn--Hilliard equation: Strong convergence rate and density convergence

Abstract

This paper presents the strong convergence rate and density convergence of a spatial finite difference method (FDM) 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 global Lipschitz nor one-sided Lipschitz. To handle this difficulty, we first utilize an interpolation approach to derive the discrete H 1 -regularity of the numerical solution. This is the key to deriving the optimal strong convergence order 1 of the numerical solution. Further, we propose a novel localization argument to estimate the total variation distance between the exact and numerical solutions, which along with the existence of the density of the numerical solution finally yields the convergence of density in L 1 (R) of the numerical solution. This partially answers positively to the open problem emerged in [9, Section 5] on computing the density of the exact solution numerically.

Strong convergence rates of an explicit scheme for stochastic Cahn-Hilliard equation with additive noise

“…To do the approximation error analysis, one often faces difficulties, raised by the presence of the unbounded operator A in front of the nonlinear term F . In the past few years, many authors investigated strong and weak approximations of stochastic Cahn-Hilliard equation [10,23,25,18,13,20,21,27,17], where some attempts to address the issue were proposed in literature. For the linearized stochastic Cahn-Hilliard equation, the readers are referred to [10,23,25].…”

Weak error estimates of fully-discrete schemes for the stochastic Cahn-Hilliard equation

“…As a consequence of ( 29) and (30), for all α P pd{2, 2szt3{2u, there exists C α P p0, 8q such that one has the inequality (31) }x} L 8 ď C α }x} α for all x P H α . In addition, for all α P pd{2, 2szt3{2u, the space H α is an algebra: there exists C α P p0, 8q such that for all x, y P H α , one has xy P H α and the inequality (32) }xy} α ď C α }x} α }y} α .…”

“…The moment bounds in the L 8 norm (first term in the left-hand side of ( 44)) are a straightforward consequence of the moment bounds in the norm } ¨}γ (second term in the left-hand side of ( 44)) and of the Sobolev inequality (31).…”

“…In the last decade, numerical analysis of such stochastic phase-field models has been a very active field of research, see, e.g., [38,44,25,37,16,27,9,4,26] and references therein. Progress on the convergence aspects of numerical approximations for stochastic phase-field model has been made recently, including strong and weak convergence rate analysis for the stochastic Allen-Cahn equation (see, e.g., [3,7,19,46,10,41,42,48,13,22,6]) and the strong convergence rate analysis for the stochastic Cahn-Hilliard equation (see, e.g., [47,20,1,21,31,14,45]).…”

Strong convergence rates of a fully discrete scheme for the Cahn-Hilliard-Cook equation

“…Although the analysis in [18,36] works quite well for CHC equation with smoother noise cases, it cannot be generalized to the rougher noise cases, especially in dimensions two and three. We would like to mention the references [1,13,17,27] on the global well-posedness and regularity estimates of CHC equation driven by the space-time white noise with d = 1. We emphasize that our spatial-temporal regularity result (1.4)- (1.5) is new in dimensions two and three.…”