Optimal-order preconditioners for the Morse-Ingard equations

TL;DR

This paper develops optimal-order preconditioners for the Morse-Ingard thermoacoustic equations, improving computational efficiency and accuracy in modeling trace gas sensors by analyzing a reformulated, weakly coupled system.

Contribution

It introduces a reformulation of the Morse-Ingard equations with weaker coupling, and constructs effective block preconditioners with proven eigenvalue bounds for faster solutions.

Findings

Preconditioners achieve optimal-order error estimates.

Eigenvalue bounds demonstrate preconditioner effectiveness.

Numerical experiments confirm theoretical results.

Abstract

The Morse-Ingard equations of thermoacoustics are a system of coupled time-harmonic equations for the temperature and pressure of an excited gas. They form a critical aspect of modeling trace gas sensors. In this paper, we analyze a reformulation of the system that has a weaker coupling between the equations than the original form. We give a G{\aa}rding-type inequality for the system that leads to optimal-order asymptotic finite element error estimates. We also develop preconditioners for the coupled system. These are derived by writing the system as a 2x2 block system with pressure and temperature unknowns segregated into separate blocks and then using either the block diagonal or block lower triangular part of this matrix as a preconditioner. Consequently, the preconditioner requires inverting smaller, Helmholtz-like systems individually for the pressure and temperature. Rigorous eigenvalue bounds are given for the preconditioned system, and these are supported by numerical experiments.