Analysis of PML Method for Stochastic Convected Helmholtz Equation

TL;DR

This paper develops and analyzes a modified PML method for stochastic convected Helmholtz equations with white noise sources, effectively truncating unbounded domains while avoiding upstream mode instabilities.

Contribution

The paper introduces a modified PML approach that eliminates upstream mode instability and removes the need for jump boundary conditions, with convergence analysis in mean-square sense.

Findings

The modified PML effectively truncates the domain without upstream mode instability.

The method provides convergence in the mean-square sense.

Analysis demonstrates the stochastic error due to domain truncation.

Abstract

We propose and analyze the perfectly matched layer (PML) method for the time-harmonic acoustic waves driven by the white noise source in the presence of the uniform flow. A PML is an artificial absorbing layer commonly used to truncate computational regions to solve problems in unbounded domains. We study a modification of PML method based on B\'ecache et. al. A truncated domain problem for stochastic convected Helmholtz equation in the infinite duct is constructed by applying PMLs. Our PML method omits the instability of inverse upstream modes in the PML. Moreover, a suitable jump condition on boundaries between computational domain and PMLs is not required. We analyze the stochastic error generated by truncations of the domain. Thus the convergence analysis of the solution is provided in the sense of mean-square.