In this paper, we consider one- and two-dimensional singularly perturbed parabolic reaction-diffusion problems. We propose a parameter-uniform numerical scheme to solve these problems. The continuous problem is first discretized in the space variable using a fitted operator finite difference method. The partial differential equation is thus transformed into a system of initial value problems which are then integrated in time with the Crank–Nicolson finite difference method. A convergence analysis shows that the scheme is second-order ε-uniform convergent in space and time. Richardson extrapolation of the space variable results in a fourth order ε-uniform convergence. Numerical experiments on two test examples confirm the theoretical findings.

在本文中，我们考虑一维和二维奇异摄动抛物线反应扩散问题。我们提出了一个参数一致的数值方案来解决这些问题。连续问题首先使用拟合算子有限差分法在空间变量中离散化。因此，偏微分方程转化为一个初始值问题系统，然后使用 Crank-Nicolson 有限差分法及时积分。收敛性分析表明，该格式在空间和时间上是二阶ε-均匀收敛的。空间变量的理查森外推导致四阶ε -均匀收敛。两个测试示例的数值实验证实了理论发现。