Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence of random splitting method for the Allen-Cahn equation in a background flow

Lei Li, Chen Wang

TL;DR

The paper analyzes the convergence of a randomized operator splitting method for the convected Allen–Cahn equation on a torus, where background incompressible flow breaks the gradient-flow structure. By establishing uniform Sobolev bounds and stability for both the PDE and the numerical scheme, the authors derive sharp local truncation error estimates and prove that the expected single-run error decays as $O(\\tau^{3/2})$ while the bias decays as $O(\\tau^{2})$ in Sobolev norms. The results rely on careful handling of unbounded nonlinear operators and Sobolev propagation, enabling rigorous error control despite the nonlinearity and convection. Numerical experiments corroborate the theory, confirming the predicted convergence rates and demonstrating practical effectiveness of the random splitting approach for phase-field convection problems.

Abstract

We study in this paper the convergence of the random splitting method for Allen-Cahn equation in a background flow that plays as a simplified model for phase separation in multiphase flows. The model does not own the gradient flow structure as the usual Allen-Cahn equation does, and the random splitting method is advantageous due to its simplicity and better convergence rate. Though the random splitting is a classical method, the analysis of the convergence is not straightforward for this model due to the nonlinearity and unboundedness of the operators. We obtain uniform estimates of various Sobolev norms of the numerical solutions and the stability of the model. Based on the Sobolev estimates, the local trunction errors are then rigorously obtained. We then prove that the random operator splitting has an expected single run error with order $1.5$ and a bias with order $2$. Numerical experiments are then performed to confirm our theoretic findings.

Convergence of random splitting method for the Allen-Cahn equation in a background flow

TL;DR

The paper analyzes the convergence of a randomized operator splitting method for the convected Allen–Cahn equation on a torus, where background incompressible flow breaks the gradient-flow structure. By establishing uniform Sobolev bounds and stability for both the PDE and the numerical scheme, the authors derive sharp local truncation error estimates and prove that the expected single-run error decays as while the bias decays as in Sobolev norms. The results rely on careful handling of unbounded nonlinear operators and Sobolev propagation, enabling rigorous error control despite the nonlinearity and convection. Numerical experiments corroborate the theory, confirming the predicted convergence rates and demonstrating practical effectiveness of the random splitting approach for phase-field convection problems.

Abstract

We study in this paper the convergence of the random splitting method for Allen-Cahn equation in a background flow that plays as a simplified model for phase separation in multiphase flows. The model does not own the gradient flow structure as the usual Allen-Cahn equation does, and the random splitting method is advantageous due to its simplicity and better convergence rate. Though the random splitting is a classical method, the analysis of the convergence is not straightforward for this model due to the nonlinearity and unboundedness of the operators. We obtain uniform estimates of various Sobolev norms of the numerical solutions and the stability of the model. Based on the Sobolev estimates, the local trunction errors are then rigorously obtained. We then prove that the random operator splitting has an expected single run error with order and a bias with order . Numerical experiments are then performed to confirm our theoretic findings.

Paper Structure

This paper contains 14 sections, 10 theorems, 170 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setup and notations
  3. The random splitting method for the model
  4. Other notations
  5. Boundedness and Stability
  6. Boundedness of Sobolev norms
  7. Stability of the Model
  8. Error Estimates of the Random Splitting Method
  9. The Local Truncation Errors
  10. Convergence analysis: expectation of the error
  11. Convergence analysis: bias
  12. Discussions
  13. Numerical Experiments
  14. Various expressions for the expansions of semigroups

Key Result

Proposition 3.1

Consider problem equ:acdrift on $\Omega$ and suppose that $v$ satisfies Assumption ass:vassumption. Then for any given $p\in[2,+\infty]$ and $k\in \mathbb{N}$, if $\|u_0\|_{k,p}+\|u_0\|_{\max(k-1,0),\infty}<\infty$, one then has

Figures (2)

  • Figure 1: The convergence order of expectation of error, for $\tau=2^{-4},2^{-5},\cdots,2^{-8}$,$N_E=10^4$. (a) $L^2$ norm; (b) $W^{1,2}$ norm.
  • Figure 2: The convergence order of bias, for $\tau=2^{-4},2^{-5},\cdots,2^{-8}$,$N_E=10^4$. (a) $L^2$ norm; (b) $W^{1,2}$ norm.

Theorems & Definitions (21)

  • Proposition 3.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.1
  • proof
  • ...and 11 more