A high-order compact LOD difference method for solving the two-dimensional diffusion reaction equation with nonlinear source term

Abstract

In this paper, a high-order compact local one-dimensional (LOD) method for the two-dimensional diffusion reaction equation with nonlinear source term is proposed. Firstly, the original equation is split into two one-dimensional equations. Then, the fourth-order compact difference formulas and the Crank–Nicolson method are used to approximate the space and time derivatives, respectively. The nonlinear source term is linearized based on the Taylor expansions. So we only need to solve two one-dimensional tri-diagonal systems at each time level, which makes the computation efficient. Through theoretical analysis, we find that the current LOD method produces a splitting error. Therefore, although this method has advantages for source term symmetry problems, it loses accuracy for source term asymmetry problems. The convergence and stability of the proposed scheme are proved by energy analysis method. Finally, several numerical examples are given to verify the accuracy and effectiveness of the proposed method. Furthermore, it is also applied in simulation of the quenching and blow-up phenomena.