Continuous and discrete parabolic operators and their qualitative properties

The basic requirement of numerical methods is convergence. However, from the practical point of view, it is generally not sufficient to construct convergent numerical methods for the solutions of partial differential equations. The qualitative adequateness of the methods is also an issue. The numerical solutions should mirror the characteristic properties of the original physical process that is modelled by the differential equation. In this paper, we give three important qualitative properties of parabolic partial differential equations: the maximum–minimum principle and its different versions, the non-negativity preservation and the maximum norm contractivity. The investigation of these properties is motivated by different physical principles. We formulate the analogues of the properties for general discrete operators and we analyse the conditions and the relations between the properties for both the continuous and the discrete operators. The approximation properties of the discrete operators are also analysed. The results of the paper are applied to the finite-difference solution methods of parabolic initial boundary-value problems.