Explicit, non-negativity-preserving and maximum-principle-satisfying finite difference scheme for the nonlinear Fisher's equation

Abstract

In this paper, a class of non-negativity-preserving and maximum-principle-satisfying finite difference methods have been derived by Vieta theorem for one-dimensional and two-dimensional Fisher's equation. By using the positivity and boundedness of numerical and exact solutions, it is shown that numerical solutions obtained by current methods converge to exact solutions with orders of O(Δt+(Δt/hx)2+hx2) for one-dimensional case and O(Δt+(Δt/hx)2+(Δt/hy)2+hx2+hy2) for two-dimensional case in the maximum norm, respectively. Here, Δt, hx and hy are meshsizes in t-, x- and y-directions, respectively. Finally, numerical results verify that the proposed method can inherit the monotonicity, boundedness and non-negativity of the continuous problems.