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

TL;DR

This paper analyzes the convergence of a random splitting numerical method for the Allen-Cahn equation with background flow, demonstrating its effectiveness and providing rigorous error estimates despite the model's nonlinearity.

Contribution

The paper provides the first rigorous convergence analysis of the random splitting method applied to a non-gradient flow Allen-Cahn model with background flow, including error order and stability results.

Findings

Expected single run error order 1.5

Bias order 2 confirmed by numerical experiments

Uniform Sobolev norm estimates for solutions

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.