Continuous and discrete parabolic operators and their qualitative properties

I. Faragó, Róbert Horváth

Research output: Contribution to journalArticle

15 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)606-631
Number of pages26
JournalIMA Journal of Numerical Analysis
Volume29
Issue number3
DOIs
Publication statusPublished - Jul 2009

Fingerprint

Parabolic Operator
Discrete Operators
Qualitative Properties
Partial differential equations
Mathematical operators
Convergence of numerical methods
Boundary value problems
Numerical methods
Mirrors
Differential equations
Numerical Methods
Contractivity
Minimum Principle
Maximum Norm
Nonnegativity
Parabolic Partial Differential Equations
Physical process
Approximation Property
Parabolic Problems
Preservation

Keywords

  • Finite-difference methods
  • Maximum principles
  • Mesh operators
  • Parabolic problems
  • Qualitative properties

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics
  • Computational Mathematics

Cite this

Continuous and discrete parabolic operators and their qualitative properties. / Faragó, I.; Horváth, Róbert.

In: IMA Journal of Numerical Analysis, Vol. 29, No. 3, 07.2009, p. 606-631.

Research output: Contribution to journalArticle

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