Qualitative properties of the numerical solution of linear parabolic problems with nonhomogeneous boundary conditions

In papers[1–3], we have formulated the conditions of the conservation of the main characteristic properties of the continuous problem to the numerical schemes in the case of homogeneous boundary conditions. Such properties are the conservation of the nonnegativity and the concavity of the initial function, monotonicity in time (contractivity) and others. In this paper, we consider similar questions to the problem with nonhomogeneous boundary conditions. We analyse the behaviour of the numerical solution in the limit by the time variable on some fixed mesh. We also consider the condition of the preservation of the nonnegativity and convexity (concavity) of the initial function. Finally, we examine the condition of the conservation of the monotonicity in the space variable of the initial function to the numerical solution.