  The second law of thermodynamics is a useful and universal tool to derive the
generalizations of the Fourier's law. In many cases, only linear relations are
considered between the thermodynamic fluxes and forces, i.e., the conduction
coefficients are independent of the temperature. In the present paper, we
investigate a particular nonlinearity in which the thermal conductivity depends
on the temperature linearly. Also, that assumption is extended to the
relaxation time, which appears in the hyperbolic generalization of Fourier's
law, namely the Maxwell-Cattaneo-Vernotte (MCV) equation. Although such
nonlinearity in the Fourier heat equation is well-known in the literature, its
extension onto the MCV equation is rarely applied. Since these nonlinearities
have significance from an experimental point of view, an efficient way is
needed to solve the system of partial differential equations. In the following,
we present a numerical method that is first developed for linear generalized
heat equations. The related stability conditions are also discussed.
