A high-order compact LOD method for solving the three-dimensional reaction–diffusion equation with nonlinear reaction term

In this paper, a high-order compact local one-dimensional (LOD) method is proposed to solve the three-dimensional reaction-diffusion equation with nonlinear reaction term. First, the governing equation is split into three one-dimensional equations. Then, the fourth-order compact difference formulas and the Crank-Nicolson method are employed to approximate the space and time derivatives, respectively. The nonlinear reaction term is linearized by utilizing the Taylor expansions. It only needs to solve three one-dimensional tridiagonal linear systems at each time layer, which makes the computation cost-effective. After conducting theoretical analysis, we find that the truncation error of the current LOD method is of order O(τ2+h4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O({\tau ^2} + {h^4})$$\end{document} for source spatially symmetry problems, but only of order O(τ+τh2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\tau + \tau {h^2})$$\end{document} for source spatially asymmetry problems. The convergence and stability of the proposed scheme are proved by the energy analysis method. Finally, some numerical experiments are carried out to demonstrate the accuracy and the efficiency of the presented method. Furthermore, our method is also exploited in the simulation of the quenching and blow-up phenomena.