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.

Compact difference schemes for heat equation with Neumann boundary conditions

“…An improved compact scheme is proposed, by which the approximate values at all boundary points can be got directly. In the second part of the article, the difference scheme given by Liao, Zhu, and Khaliq in [10] is discussed and stability and convergence with the convergence order O(τ 2 + h 3.5 ) in a discrete maximum norm are obtained. Both the advantages and disadvantages of the difference scheme are pointed out.…”

“…Omitting the small terms R n+ 1 2 i in (3.1), µ n in (3.4), ν n in (3.6), we obtain the difference scheme (1.8)-(1.11) presented by Liao, Zhu, and Khaliq in [10]. Denoteα…”

“…Liao, Zhu, and Khaliq [10] proposed a high order accuracy difference scheme for nonlinear reaction-diffusion equations with Neumann boundary conditions. When applied to the linear equation Numerical examples were presented to demonstrate the accuracy and efficiency of the scheme.…”

Compact difference schemes for heat equation with Neumann boundary conditions (II)

“…This fact may result in the decreasing of global convergence order of difference scheme. Sun [8] analyzed two difference schemes: (1.8)-(1.11) proposed by Zhao et al [10] and (1.4)-(1.7) developed by Liao et al [9]. Numerical convergence order for (1.8)-(1.11) is O(τ 2 + h 3 ), whereas theoretical convergence order in [8] 3.5 ).…”

“…One of them, proposed by Liao et al [9], was derived by introducing two fictitious points at each temporal level, which was expressed after eliminating these fictitious values by …”