In this paper, we constructed a fitted mesh finite difference method for solving a class of time-dependent singularly perturbed turning point convection-diffusion problems whose solution exhibits an interior layer. The diffusion coefficient in the underlying PDE is a quadratic function of the space variable and contains a perturbation parameter. While such problems have been studied in the case of boundary layers, little has been achieved for interior layer problems where the coefficient functions are considered to be dependent on the space variable alone. In this work, we focus our attention to such problems where the coefficient functions are dependent of both the space and time variables. Following the work of Liseikin (USSR Computational Mathematics and Mathematical Physics 26(6), 133–139, 1986), we establish bounds on the solution and its derivatives. Then we discretize the time derivative using an implicit Euler method. This discretization results in a set of two-point boundary value problems (TPBVPs). We then construct a fitted mesh finite difference method to solve these TPBVPs. This method is analyzed for stability and convergence. We proved that it satisfies a minimum principle and is uniformly convergent with respect to the perturbation parameter. In order to improve the accuracy of the proposed method, we use the Richardson extrapolation. Finally, we present some numerical experiments to validate our theoretical findings.

在本文中，我们构建了一种拟合网格有限差分方法，用于求解一类具有内层的瞬态奇异摄动转折点对流扩散问题。底层 PDE 中的扩散系数是空间变量的二次函数，包含一个扰动参数。虽然已经在边界层的情况下研究了此类问题，但对于系数函数被认为仅依赖于空间变量的内层问题，几乎没有取得什么成果。在这项工作中，我们将注意力集中在系数函数依赖于空间和时间变量的问题上。根据 Liseikin 的工作（苏联计算数学和数学物理 26(6), 133–139, 1986），我们建立解决方案及其导数的界限。然后我们使用隐式欧拉方法离散时间导数。这种离散化导致了一组两点边值问题 (TPBVP)。然后我们构建了一个拟合网格有限差分方法来解决这些 TPBVP。分析该方法的稳定性和收敛性。我们证明了它满足最小原则并且对于扰动参数是一致收敛的。为了提高所提出方法的准确性，我们使用了理查森外推法。最后，我们提出了一些数值实验来验证我们的理论发现。然后我们构建了一个拟合网格有限差分方法来解决这些 TPBVP。分析该方法的稳定性和收敛性。我们证明了它满足最小原则并且对于扰动参数是一致收敛的。为了提高所提出方法的准确性，我们使用了理查森外推法。最后，我们提出了一些数值实验来验证我们的理论发现。然后我们构建了一个拟合网格有限差分方法来解决这些 TPBVP。分析该方法的稳定性和收敛性。我们证明了它满足最小原则并且对于扰动参数是一致收敛的。为了提高所提出方法的准确性，我们使用了理查森外推法。最后，我们提出了一些数值实验来验证我们的理论发现。