Fully Implicit Finite Volume Scheme for Transient Conductive-Radiative Heat Transfer: Discrete Existence, Uniqueness, Maximum Principle

TL;DR

This paper develops a fully implicit finite volume scheme for transient nonlinear heat transfer with diffuse-gray radiation, proving existence, uniqueness, and maximum principles in three dimensions.

Contribution

It extends previous semi-implicit schemes to a fully implicit scheme, establishing discrete maximum principles and solution uniqueness for complex 3D models.

Findings

Proves discrete maximum principle for the scheme

Establishes existence and uniqueness of solutions

Provides a priori bounds for the numerical solutions

Abstract

This article studies a fully implicit finite volume scheme for transient nonlinear heat transport equations coupled by nonlocal interface conditions modeling diffuse-gray radiation between the surfaces of (both open and closed) cavities. The model is considered in three space dimensions; modifications for the axisymmetric case are indicated. Extending previous results, where a similar, but not fully implicit, finite volume scheme was considered, a discrete maximum principle is established, yielding discrete $L^\infty$-$L^\infty$ a priori bounds as well as a unique discrete solution to the finite volume scheme.