Discrete maximum principle and adequate discretizations of linear parabolic problems

I. Faragó, Róbert Horváth

Research output: Article

49 Citations (Scopus)

Abstract

In this paper, we analyze the connections between the different qualitative properties of numerical solutions of linear parabolic problems with Dirichlet-type boundary condition. First we formulate the qualitative properties for the differential equations and shed light on their relations. Then we show how the well-known discretization schemes can be written in the form of a one-step iterative process. We give necessary and sufficient conditions of the main qualitative properties of these iterations. We apply the results to the finite difference and Galerkin finite element solutions of linear parabolic problems. In our main result we show that the nonnegativity preservation property is equivalent to the maximum-minimum principle and they imply the maximum norm contractivity. In one, two, and three dimensions, we list sufficient a priori conditions that ensure the required qualitative properties. Finally, we demonstrate the above results on numerical examples.

Original languageEnglish
Pages (from-to)2313-2336
Number of pages24
JournalSIAM Journal on Scientific Computing
Volume28
Issue number6
DOIs
Publication statusPublished - 2006

Fingerprint

Discrete Maximum Principle
Maximum principle
Qualitative Properties
Parabolic Problems
Differential equations
Discretization
Boundary conditions
Contractivity
Minimum Principle
Maximum Norm
Discretization Scheme
Nonnegativity
Finite Element Solution
Iterative Process
Galerkin
Preservation
One Dimension
Dirichlet
Three-dimension
Finite Difference

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Discrete maximum principle and adequate discretizations of linear parabolic problems. / Faragó, I.; Horváth, Róbert.

In: SIAM Journal on Scientific Computing, Vol. 28, No. 6, 2006, p. 2313-2336.

Research output: Article

@article{8cd8200b38ab440c8ac77a3f5b34184a,
title = "Discrete maximum principle and adequate discretizations of linear parabolic problems",
abstract = "In this paper, we analyze the connections between the different qualitative properties of numerical solutions of linear parabolic problems with Dirichlet-type boundary condition. First we formulate the qualitative properties for the differential equations and shed light on their relations. Then we show how the well-known discretization schemes can be written in the form of a one-step iterative process. We give necessary and sufficient conditions of the main qualitative properties of these iterations. We apply the results to the finite difference and Galerkin finite element solutions of linear parabolic problems. In our main result we show that the nonnegativity preservation property is equivalent to the maximum-minimum principle and they imply the maximum norm contractivity. In one, two, and three dimensions, we list sufficient a priori conditions that ensure the required qualitative properties. Finally, we demonstrate the above results on numerical examples.",
keywords = "Contractivity in maximum norm, Discrete maximum principle, Heat conduction, Nonnegativity, Numerical solution, Parabolic problems, Qualitative properties",
author = "I. Farag{\'o} and R{\'o}bert Horv{\'a}th",
year = "2006",
doi = "10.1137/050627241",
language = "English",
volume = "28",
pages = "2313--2336",
journal = "SIAM Journal of Scientific Computing",
issn = "1064-8275",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "6",

}

TY - JOUR

T1 - Discrete maximum principle and adequate discretizations of linear parabolic problems

AU - Faragó, I.

AU - Horváth, Róbert

PY - 2006

Y1 - 2006

N2 - In this paper, we analyze the connections between the different qualitative properties of numerical solutions of linear parabolic problems with Dirichlet-type boundary condition. First we formulate the qualitative properties for the differential equations and shed light on their relations. Then we show how the well-known discretization schemes can be written in the form of a one-step iterative process. We give necessary and sufficient conditions of the main qualitative properties of these iterations. We apply the results to the finite difference and Galerkin finite element solutions of linear parabolic problems. In our main result we show that the nonnegativity preservation property is equivalent to the maximum-minimum principle and they imply the maximum norm contractivity. In one, two, and three dimensions, we list sufficient a priori conditions that ensure the required qualitative properties. Finally, we demonstrate the above results on numerical examples.

AB - In this paper, we analyze the connections between the different qualitative properties of numerical solutions of linear parabolic problems with Dirichlet-type boundary condition. First we formulate the qualitative properties for the differential equations and shed light on their relations. Then we show how the well-known discretization schemes can be written in the form of a one-step iterative process. We give necessary and sufficient conditions of the main qualitative properties of these iterations. We apply the results to the finite difference and Galerkin finite element solutions of linear parabolic problems. In our main result we show that the nonnegativity preservation property is equivalent to the maximum-minimum principle and they imply the maximum norm contractivity. In one, two, and three dimensions, we list sufficient a priori conditions that ensure the required qualitative properties. Finally, we demonstrate the above results on numerical examples.

KW - Contractivity in maximum norm

KW - Discrete maximum principle

KW - Heat conduction

KW - Nonnegativity

KW - Numerical solution

KW - Parabolic problems

KW - Qualitative properties

UR - http://www.scopus.com/inward/record.url?scp=36349001263&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=36349001263&partnerID=8YFLogxK

U2 - 10.1137/050627241

DO - 10.1137/050627241

M3 - Article

AN - SCOPUS:36349001263

VL - 28

SP - 2313

EP - 2336

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

SN - 1064-8275

IS - 6

ER -