Grid equidistribution for reaction–diffusion problems in one dimension

Abstract

The numerical solution of a linear singularly-perturbed reaction-diffusion two-point boundary value problem is considered. The method used is adaptive movement of a fixed number of mesh points by monitor-function equidistribution. A partly heuristic argument based on truncation error analysis leads to several suitable monitor functions, but also shows that the standard arc-length monitor function is unsuitable for this problem. Numerical results are provided to demonstrate the effectiveness of our preferred monitor function.

Optimal error estimate using mesh equidistribution technique for singularly perturbed system of reaction–diffusion boundary-value problems

“…Lemma 2.1 [12]. Suppose the coefficients of the following reaction-diffusion problem b Lv Àev 00 ðxÞ þ bðxÞvðxÞ ¼ f ðxÞ; x 2 X; vð0Þ ¼ 0; vð1Þ ¼ 0;…”

“…It can also be seen from [12] that the discrete solution v of (3.3) satisfies the following inequality jjvjj X N 6 Ce À1=2 jjf jj À1;1 : ð3:5Þ …”

“…For further analysis, we shall use the procedure provided in [12] used for the scalar reaction-diffusion problem. For i ¼ 1ð1ÞN À 1, a straight forward calculation provided in [12] shows that e m ðd 2 u m À u 00…”

“…A monitor function based on the arc-length of the solution, is analyzed by several authors, for example, Qiu et al [25] for convection-diffusion problems. However, Kopteva et al [12] pointed out that the mesh equidistribution based on arc-length monitor function is suitable only for convection-diffusion type problems and not for reaction-diffusion type problems. For scalar reaction-diffusion problems, a new monitor function is proposed in [11], which is very similar to the one, obtained by Beckett and Mackenzie [1].…”

Numerical Solution Of Linear Emden Fowler Boundary Value Problem In Fuzzy Environment

“…Thus, considering their approximate solutions is becoming more important. The two-point boundary value problem (TPBVP) occurs in a wide variety of problems in engineering [3,4] and science, including the modeling of chemical reactions [5,6], heat transfer [7,8], and diffusion [9,10], and the solution of optimal control problems [11,12]. Fuzzy two point boundary value problems (FTPBVP) appears when the modeling of these problems cannot be sure is perfect and its nature is under uncertainty.…”