A Fourth-order Compact Algorithm for Nonlinear Reaction-diffusion Equations with Neumann Boundary Conditions

Wenyuan Liao; Jianping Zhu; Abdul Q.M. Khaliq

doi:10.1002/num.20111

A Fourth-order Compact Algorithm for Nonlinear Reaction-diffusion Equations with Neumann Boundary Conditions

Wenyuan Liao
, Jianping Zhu
, Abdul Q.M. Khaliq

University of Calgary
Middle Tennessee State University

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

Abstract

In this article, we discuss a scheme for dealing with Neumann and mixed boundary conditions using a compact stencil. The resulting compact algorithm for solving systems of nonlinear reaction-diffusion equations is fourth-order accurate in both the temporal and spatial dimensions. We also prove that the standard second-order approximation to zero Neumann boundary conditions provides fourth-order accuracy when the nonlinear reaction term is independent of the spatial variables. Numerical examples, including an application of this algorithm to a mathematical model describing frontal polymerization process, are presented in the article to demonstrate the accuracy and efficiency of the scheme.

Original language	American English
Journal	Numerical Methods for Partial Differential Equations
Volume	22
DOIs	https://doi.org/10.1002/num.20111
State	Published - Jul 26 2005

Keywords

Reaction-diffusion equation
Neumann boundary condition
Compact algorithms

Disciplines

Partial Differential Equations

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10.1002/num.20111License: CC BY

http://onlinelibrary.wiley.com/doi/10.1002/num.20111/abstract

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title = "A Fourth-order Compact Algorithm for Nonlinear Reaction-diffusion Equations with Neumann Boundary Conditions",

abstract = " In this article, we discuss a scheme for dealing with Neumann and mixed boundary conditions using a compact stencil. The resulting compact algorithm for solving systems of nonlinear reaction-diffusion equations is fourth-order accurate in both the temporal and spatial dimensions. We also prove that the standard second-order approximation to zero Neumann boundary conditions provides fourth-order accuracy when the nonlinear reaction term is independent of the spatial variables. Numerical examples, including an application of this algorithm to a mathematical model describing frontal polymerization process, are presented in the article to demonstrate the accuracy and efficiency of the scheme.",

keywords = "Reaction-diffusion equation, Neumann boundary condition, Compact algorithms",

author = "Wenyuan Liao and Jianping Zhu and Khaliq, \{Abdul Q.M.\}",

note = "Liao, W., Zhu, J., and Khaliq, A. Q. M. (2006). A Fourth-order Compact Algorithm for Nonlinear Reaction-diffusion Equations with Neumann Boundary Conditions. Numerical Methods for Partial Differential Equations, 22(3), 600 - 616, doi: 10.1002/num.20111.",

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T1 - A Fourth-order Compact Algorithm for Nonlinear Reaction-diffusion Equations with Neumann Boundary Conditions

AU - Liao, Wenyuan

AU - Zhu, Jianping

AU - Khaliq, Abdul Q.M.

N1 - Liao, W., Zhu, J., and Khaliq, A. Q. M. (2006). A Fourth-order Compact Algorithm for Nonlinear Reaction-diffusion Equations with Neumann Boundary Conditions. Numerical Methods for Partial Differential Equations, 22(3), 600 - 616, doi: 10.1002/num.20111.

PY - 2005/7/26

Y1 - 2005/7/26

N2 - In this article, we discuss a scheme for dealing with Neumann and mixed boundary conditions using a compact stencil. The resulting compact algorithm for solving systems of nonlinear reaction-diffusion equations is fourth-order accurate in both the temporal and spatial dimensions. We also prove that the standard second-order approximation to zero Neumann boundary conditions provides fourth-order accuracy when the nonlinear reaction term is independent of the spatial variables. Numerical examples, including an application of this algorithm to a mathematical model describing frontal polymerization process, are presented in the article to demonstrate the accuracy and efficiency of the scheme.

AB - In this article, we discuss a scheme for dealing with Neumann and mixed boundary conditions using a compact stencil. The resulting compact algorithm for solving systems of nonlinear reaction-diffusion equations is fourth-order accurate in both the temporal and spatial dimensions. We also prove that the standard second-order approximation to zero Neumann boundary conditions provides fourth-order accuracy when the nonlinear reaction term is independent of the spatial variables. Numerical examples, including an application of this algorithm to a mathematical model describing frontal polymerization process, are presented in the article to demonstrate the accuracy and efficiency of the scheme.

KW - Reaction-diffusion equation

KW - Neumann boundary condition

KW - Compact algorithms

UR - https://engagedscholarship.csuohio.edu/scimath_facpub/55

UR - http://onlinelibrary.wiley.com/doi/10.1002/num.20111/abstract

U2 - 10.1002/num.20111

DO - 10.1002/num.20111

M3 - Article

VL - 22

JO - Numerical Methods for Partial Differential Equations

JF - Numerical Methods for Partial Differential Equations

ER -

A Fourth-order Compact Algorithm for Nonlinear Reaction-diffusion Equations with Neumann Boundary Conditions

Abstract

Keywords

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