中文摘要：

利用二阶微商的三次样条四阶紧致差分逼近公式，推导出两种数值求解二维扩散反应方程的两层9点加权隐式紧致差分格式．当θ=1／2时，该格式在时间和空间方向上分别达到二阶和四阶精度．通过Fourier方法讨论知，当1／2≤θ≤1时，格式是无条件稳定的；当0≤θ〈1／2时，格式是条件稳定的．为了克服传统迭代法在求解隐格式方面的困难，差分方程采用多重网格方法进行求解并将本文格式的结果与P-R格式及C-N格式下的结果进行比较．数值实验结果验证本文方法的精确性和可靠性及多重网格方法的效率．

英文摘要：

Applying second derivative approximation formula with cubic spline and fourth-order compact difference, two kinds of weighted implicit compact difference formats with two levels and nine points were derived for solving two-dimensional diffusion-reaction equation. The two formats could achieve second-or- der accuracy in time and fourth-order accuracy in space respectively when θ= 1/2. By means of the Fourier analysis, it could be concluded that when 1/2≤θ≤1, they would be unconditionally stable. When 0≤θ〈1/2, they would be conditional stable. In order to overcome the ciifficulty of conventional method in sol- ving the implicit format, the multigrid method was adopted to acc1crate convergence rate. The numerical result from two formats presented were compared with that from P-R and C-N formats. The comparison result verified the accuracy and stability of two schemes presented and the efficiency of multigrid method.