Unconditionally monotone and globally stable difference schemes for the Fisher equation

Abstract

In this paper, we construct and study unconditionally monotone and globally stable difference schemes for the Fisher equation. It has been shown that constructed schemes inherit the stability property of the exact solution: 0 ≤ u(x, t) ≤ 1, (x, t) ∈ QT = {(x, t) : 0 ≤ x ≤ l, 0 ≤ t < +∞} for a given input data of the problem. The unconditional monotonicity of the difference schemes is proved and the a priori estimate is obtained in the uniform norm for the difference solution. The stable behavior of the difference solution in the nonlinear case takes place under slightly more stringent constraints on the input data: 0,5 ≤ u0 (x), µ1(t), µ2(t) ≤ 1.