Some sufficient conditions for the non-negativity preservation property in the discrete heat conduction model

In the course of the numerical approximation of mathematical models there is often a need to solve a system of linear equations with a tridiagonal or a block-tridiagonal matrices. Usually it is efficient to solve these systems using a special algorithm (tridiagonal matrix algorithm or TDMA) which takes advantage of the structure. The main result of this work is to formulate a sufficient condition for the numerical method to preserve the non-negativity for the special algorithm for structured meshes. We show that a different condition can be obtained for such cases where there is no way to fulfill this condition. Moreover, as an example, the numerical solution of the two-dimensional heat conduction equation on a rectangular domain is investigated by applying Dirichlet boundary condition and Neumann boundary condition on different parts of the boundary of the domain. For space discretization, we apply the linear finite element method, and for time discretization, the well-known Θ-method. The theoretical results of the paper are verified by several numerical experiments.