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.
A Fourth-order Compact Algorithm for Nonlinear Reaction-diffusion Equations with Neumann Boundary ConditionsNumerical Methods for Partial Differential Equations
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Citation InformationLiao, 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.