Energy regularized models for logarithmic SPDEs and their numerical approximations

TL;DR

This paper introduces energy regularized models for logarithmic SPDEs, ensuring stability and robustness, and develops numerical schemes to accurately approximate their behavior, with applications in phase separation and soft matter modeling.

Contribution

It proposes new regularized models for logarithmic SPDEs that preserve energy laws and maximum bounds, along with semi-implicit numerical methods for their approximation.

Findings

Models recover classical dynamics in the deterministic limit.

Maximum principle holds for multiplicative noise.

Numerical results verify theoretical properties.

Abstract

Understanding the properties of the stochastic phase field models is crucial to model processes in several practical applications, such as soft matters and phase separation in random environments. To describe such random evolution, this work proposes and studies two mathematical models and their numerical approximations for parabolic stochastic partial differential equation (SPDE) with a logarithmic Flory--Huggins energy potential. These multiscale models are built based on a regularized energy technique and thus avoid possible singularities of coefficients. According to the large deviation principle, we show that the limit of the proposed models with small noise naturally recovers the classical dynamics in deterministic case. Moreover, when the driving noise is multiplicative, the Stampacchia maximum principle holds which indicates the robustness of the proposed model. One of the main advantages of the proposed models is that they can admit the energy evolution law and asymptotically preserve the Stampacchia maximum bound of the original problem. To numerically capture these asymptotic behaviors, we investigate the semi-implicit discretizations for regularized logrithmic SPDEs. Several numerical results are presented to verify our theoretical findings.