## 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.