Analysis of a mixed space-time diffusion equation

Ebrahim Momoniat

doi:10.1007/s00033-014-0433-6

Analysis of a mixed space-time diffusion equation

Ebrahim Momoniat

University of the Witwatersrand

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

An energy method is used to analyze the stability of solutions of a mixed space-time diffusion equation that has application in the unidirectional flow of a second-grade fluid and the distribution of a compound Poisson process. Solutions to the model equation satisfying Dirichlet boundary conditions are proven to dissipate total energy and are hence stable. The stability of asymptotic solutions satisfying Neumann boundary conditions coincides with the condition for the positivity of numerical solutions of the model equation from a Crank–Nicolson scheme. The Crank–Nicolson scheme is proven to yield stable numerical solutions for both Dirichlet and Neumann boundary conditions for positive values of the critical parameter. Numerical solutions are compared to analytical solutions that are valid on a finite domain.

Original language	English
Pages (from-to)	1175-1186
Number of pages	12
Journal	Zeitschrift fur Angewandte Mathematik und Physik
Volume	66
Issue number	3
DOIs	https://doi.org/10.1007/s00033-014-0433-6
Publication status	Published - 28 Jun 2015
Externally published	Yes

Keywords

Crank–Nicolson scheme
Dissipation of total energy
Energy method
Stable numerical method

ASJC Scopus subject areas

General Mathematics
General Physics and Astronomy
Applied Mathematics

Access to Document

10.1007/s00033-014-0433-6

Cite this

@article{52adc6866dcb4725846251faa52f4a34,

title = "Analysis of a mixed space-time diffusion equation",

abstract = "An energy method is used to analyze the stability of solutions of a mixed space-time diffusion equation that has application in the unidirectional flow of a second-grade fluid and the distribution of a compound Poisson process. Solutions to the model equation satisfying Dirichlet boundary conditions are proven to dissipate total energy and are hence stable. The stability of asymptotic solutions satisfying Neumann boundary conditions coincides with the condition for the positivity of numerical solutions of the model equation from a Crank–Nicolson scheme. The Crank–Nicolson scheme is proven to yield stable numerical solutions for both Dirichlet and Neumann boundary conditions for positive values of the critical parameter. Numerical solutions are compared to analytical solutions that are valid on a finite domain.",

keywords = "Crank–Nicolson scheme, Dissipation of total energy, Energy method, Stable numerical method",

author = "Ebrahim Momoniat",

note = "Publisher Copyright: {\textcopyright} 2014, Springer Basel.",

year = "2015",

month = jun,

day = "28",

doi = "10.1007/s00033-014-0433-6",

language = "English",

volume = "66",

pages = "1175--1186",

journal = "Zeitschrift fur Angewandte Mathematik und Physik",

issn = "0044-2275",

publisher = "Birkhauser Verlag Basel",

number = "3",

}

TY - JOUR

T1 - Analysis of a mixed space-time diffusion equation

AU - Momoniat, Ebrahim

PY - 2015/6/28

Y1 - 2015/6/28

N2 - An energy method is used to analyze the stability of solutions of a mixed space-time diffusion equation that has application in the unidirectional flow of a second-grade fluid and the distribution of a compound Poisson process. Solutions to the model equation satisfying Dirichlet boundary conditions are proven to dissipate total energy and are hence stable. The stability of asymptotic solutions satisfying Neumann boundary conditions coincides with the condition for the positivity of numerical solutions of the model equation from a Crank–Nicolson scheme. The Crank–Nicolson scheme is proven to yield stable numerical solutions for both Dirichlet and Neumann boundary conditions for positive values of the critical parameter. Numerical solutions are compared to analytical solutions that are valid on a finite domain.

AB - An energy method is used to analyze the stability of solutions of a mixed space-time diffusion equation that has application in the unidirectional flow of a second-grade fluid and the distribution of a compound Poisson process. Solutions to the model equation satisfying Dirichlet boundary conditions are proven to dissipate total energy and are hence stable. The stability of asymptotic solutions satisfying Neumann boundary conditions coincides with the condition for the positivity of numerical solutions of the model equation from a Crank–Nicolson scheme. The Crank–Nicolson scheme is proven to yield stable numerical solutions for both Dirichlet and Neumann boundary conditions for positive values of the critical parameter. Numerical solutions are compared to analytical solutions that are valid on a finite domain.

KW - Crank–Nicolson scheme

KW - Dissipation of total energy

KW - Energy method

KW - Stable numerical method

UR - http://www.scopus.com/inward/record.url?scp=84938291082&partnerID=8YFLogxK

U2 - 10.1007/s00033-014-0433-6

DO - 10.1007/s00033-014-0433-6

M3 - Article

AN - SCOPUS:84938291082

SN - 0044-2275

VL - 66

SP - 1175

EP - 1186

JO - Zeitschrift fur Angewandte Mathematik und Physik

JF - Zeitschrift fur Angewandte Mathematik und Physik

IS - 3

ER -

Analysis of a mixed space-time diffusion equation

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this