On the monotonicity conservation in numerical solutions of the heat equation Original Research Article

It is important to choose such numerical methods in practice that mirror the characteristic properties of the described process beyond the stability and convergence. The investigated qualitative property in this paper is the conservation of the monotonicity in space of the initial heat distribution. We prove some statements about the monotonicity conservation and total monotonicity of one-step vector-iterations. Then, applying these results, we consider the numerical solutions of the one-dimensional heat equation. Our main theorem formulates the necessary and sufficient condition of the uniform monotonicity conservation. The sharpness of the conditions is demonstrated by numerical examples.