Justification of a discrete analog of the conjugate-operator model of the heat conduction problem

For the conjugate-operator model of the heat conduction problem, we construct and justify a discrete analog preserving the structure of the initial model. The justification of convergence is carried out for a difference scheme in the conjugate-operator form. It is shown that the difference scheme converges with second-order accuracy for the cases of discontinuous medium parameters in the Fourier law and nonuniform grids.