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IMA Journal of Numerical Analysis Advance Access originally published online on June 16, 2008
IMA Journal of Numerical Analysis 2009 29(3):606-631; doi:10.1093/imanum/drn032
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Continuous and discrete parabolic operators and their qualitative properties

István Faragó{dagger}

Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University, Pázmány P. Sétány I/C, Budapest H-1117, Hungary

Róbert Horváth{ddagger}

Institute of Mathematics and Statistics, University of West-Hungary, Erzsébet u. 9, Sopron H-9400, Hungary

{dagger} Corresponding author. Email: faragois{at}cs.elte.hu

{ddagger} Email: rhorvath{at}ktk.nyme.hu

Received on 22 November 2007. Revised on 29 April 2008.


  Abstract

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.

Key Words: mesh operators; finite-difference methods; parabolic problems; qualitative properties; maximum principles


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