Integral Equation Methods for the Morse-Ingard Equations

TL;DR

This paper introduces two integral-equation-based methods for solving the Morse-Ingard equations, demonstrating their accuracy and efficiency in complex geometries in both 2D and 3D.

Contribution

The paper develops and compares a decoupled and a coupled second-kind integral equation method for Morse-Ingard equations, highlighting their respective advantages and limitations.

Findings

Coupled method is well-conditioned and highly accurate.

Decoupled method offers lower computational cost and flexibility.

Numerical examples show effectiveness in 2D and 3D complex geometries.

Abstract

We present two (a decoupled and a coupled) integral-equation-based methods for the Morse-Ingard equations subject to Neumann boundary conditions on the exterior domain. Both methods are based on second-kind integral equation (SKIE) formulations. The coupled method is well-conditioned and can achieve high accuracy. The decoupled method has lower computational cost and more flexibility in dealing with the boundary layer; however, it is prone to the ill-conditioning of the decoupling transform and cannot achieve as high accuracy as the coupled method. We show numerical examples using a Nystr\"om method based on quadrature-by-expansion (QBX) with fast-multipole acceleration. We demonstrate the accuracy and efficiency of the solvers in both two and three dimensions with complex geometry.