## A Finite Volume Scheme for Transient Nonlocal Conductive-Radiative Heat Transfer, Part 1: Formulation and Discrete Maximum Principle

#### Peter Philip, IMA

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A finite volume scheme for transient nonlinear heat transport equations coupled by nonlocal interface conditions is formulated, and the discrete is analyzed. The interface conditions model diffuse-gray radiation between the surfaces of (both open and closed) cavities. The model is considered in three space dimensions. The special difficulties of the problem lie in the radiative nonlocal coupling between surfaces and in the allowed nonlinear dependence of internal energy and emissivities on the solution (i.e. temperature). Moreover, at material interfaces, the internal energy and the (otherwise constant) diffusion coefficient can be discontinuous. A discrete maximum principle for the finite volume scheme is established, yielding discrete $L^\infty$-$L^\infty$ a priori bounds as well as a unique discrete solution to the finite volume scheme.