In this paper, high-order compact difference method is used to solve the one-dimensional nonlinear advection diffusion reaction equation. The nonlinearity here is mainly reflected in the advection and reaction terms. Firstly, the diffusion term is discretized by using the fourth-order compact difference formula, the nonlinear advection term is approximated by using the fourth-order Padé formula of the first-order derivative, and the time derivative term is discretized by using the fourth-order backward differencing formula. An unconditionally stable five-step fourth-order fully implicit compact difference scheme is developed. This scheme has fourth-order accuracy in both time and space. Secondly, for the calculations of the start-up time steps, the time derivative term is discretized by the Crank-Nicolson method, and Richardson extrapolation formula is used to improve the accuracy in time direction from the second-order to the fourth-order. Thirdly, convergence and stability of the difference scheme in $H^1$ seminorm, $L^{\infty }$ and $L^2$ norms, existence and uniqueness of the numerical solutions are proved, respectively. Fourthly, the Thomas algorithm is used to solve the nonlinear algebraic equations at each time step, and a time advancement algorithm with linearized iteration strategy is established. Finally, the accuracy, stability and efficiency of the present approach are verified by some numerical experiments.

本文采用高阶紧致差分法求解一维非线性对流扩散反应方程。这里的非线性主要反映在对流和反应项中。首先，使用四阶紧致差分公式将扩散项离散化，使用四阶Padé逼近非线性对流项一阶导数公式，时间导数项通过使用四阶后向差分公式离散化。提出了一种无条件稳定的五步四阶完全隐式紧致差分格式。该方案在时间和空间上均具有四阶精度。其次，对于启动时间步长的计算，采用Crank-Nicolson方法离散时间导数项，并采用Richardson外推公式来提高时间方向上从二阶到四阶的精度。第三，差分方案的收敛性和稳定性$H^{\mathrm{1个}}$ 半范式 ${\mathrm{大号}}^{\infty }$ 和 ${\mathrm{大号}}^{\mathrm{2个}}$分别证明了数值解的范数，存在性和唯一性。第四，采用托马斯算法求解每个时间步长的非线性代数方程，建立了具有线性化迭代策略的时间提前算法。最后，通过一些数值实验验证了该方法的准确性，稳定性和有效性。