Variational approach to the construction of finite-difference schemes for the heat equation on curvilinear networks

In the present paper, it is proposed to construct difference schemes, written in terms of heat fluxes, for the heat equation, using curvilinear networks. The construction of the schemes is based on a variational principle for the heat fluxes, formulated in this paper. The operators of the difference schemes retain such important properties as self-adjointness and the property of having fixed sign. The schemes are stable and provide second-order approximation with respect to space and first-order approximation with respect to time.