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A scalar auxiliary variable-based semi-implicit scheme for stochastic Cahn--Hilliard equation

Jianbo Cui, Jie Shen, Derui Sheng, Yahong Xiang

Abstract

In this paper, we present a novel semi-implicit numerical scheme for the stochastic Cahn--Hilliard equation driven by multiplicative noise. By reformulating the original equation into an equivalent stochastic scalar auxiliary variable (SSAV) system, our method enables an efficient and stable treatment of polynomial nonlinearities in a semi-implicit fashion. In order to accurately capture the impact of stochastic perturbations, we carefully incorporate Itô correction terms into the SSAV approximation. Leveraging the smoothing properties of the underlying semigroup and the $H^{-1}$-dissipative structure of the nonlinear term, we establish the optimal strong convergence order of one-half for the proposed scheme in the trace-class noise case. Moreover, we show that the modified SAV energy asymptotically preserves the energy evolution law. Finally, numerical experiments are provided to validate the theoretical results and to explore the influence of noise near the sharp-interface limit.

A scalar auxiliary variable-based semi-implicit scheme for stochastic Cahn--Hilliard equation

Abstract

In this paper, we present a novel semi-implicit numerical scheme for the stochastic Cahn--Hilliard equation driven by multiplicative noise. By reformulating the original equation into an equivalent stochastic scalar auxiliary variable (SSAV) system, our method enables an efficient and stable treatment of polynomial nonlinearities in a semi-implicit fashion. In order to accurately capture the impact of stochastic perturbations, we carefully incorporate Itô correction terms into the SSAV approximation. Leveraging the smoothing properties of the underlying semigroup and the -dissipative structure of the nonlinear term, we establish the optimal strong convergence order of one-half for the proposed scheme in the trace-class noise case. Moreover, we show that the modified SAV energy asymptotically preserves the energy evolution law. Finally, numerical experiments are provided to validate the theoretical results and to explore the influence of noise near the sharp-interface limit.
Paper Structure (25 sections, 16 theorems, 221 equations, 5 figures, 2 tables)

This paper contains 25 sections, 16 theorems, 221 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Preliminaries
  4. Main results
  5. Exponential Euler SSAV scheme
  6. Regularity estimates
  7. Spatial regularity estimate
  8. Temporal regularity estimate
  9. Proof of Theorem \ref{['Thm3_main']}
  10. Error estimate between $\Phi$ and $X$
  11. Error estimate between $\phi$ and $\Phi$
  12. Proof of Theorem \ref{['Sec:energy:pro']}
  13. Numerical experiments
  14. Averaged energy evolution
  15. Sharp-interface dynamics
  16. ...and 10 more sections

Key Result

Proposition 1

Let Assumptions assum2 and assum1 hold, and let $\phi^0\in \dot{H}^{\beta}(\mathcal{O})$ for some $\beta\in[1,2)$. Then for any $p\ge 1$, there exists a constant $C:=C(p,\beta)>0$ such that

Figures (5)

  • Figure 1: Comparison of averaged energies of the exponential Euler SSAV scheme and standard SAV scheme for equation \ref{['ex:eps=1']}.
  • Figure 2: Individual snapshots of the zero--level set of the solution at several time points for equation \ref{['model_epsilon_gamma']} with $\gamma=1$.
  • Figure 3: Snapshots of the zero--level set of the solution at several time points for deterministic Cahn--Hilliard equation.
  • Figure 4: Individual snapshots of the zero--level set of the solution at several time points for equation \ref{['model_epsilon_gamma']} with $\gamma=0$.
  • Figure 5: Averaged snapshots of the zero--level set of the solution at several time points for equation \ref{['model_epsilon_gamma']} with $\gamma=0$.

Theorems & Definitions (30)

  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Remark 2
  • Lemma 4
  • proof
  • Corollary 5
  • Lemma 6
  • Lemma 7
  • ...and 20 more