In the present paper, we describe cubic spline combined with high accuracy compact formulation for computing approximate solution values to nonlinear convection-dominated diffusion equations, and apply the method to several variants of Burger’s type parabolic partial differential equations. A combination of cubic spline interpolating polynomial and quasi-variable grids sequence yields a two-level implicit compact computational scheme that falls in the range of increased accuracy. The grid stretching parameter plays an important role while considering the boundary layer phenomenon to convection-dominated parabolic equations. The discretization on the quasi-variable grid network yields a more accurate and precise solution than those obtained on a uniform grid network. It is easy to extend the proposed scheme to the singular equation with compact operators, preserving the order and accuracy. It is shown through stability analysis that the new method is stable with an accuracy of order four and two in the spatial and temporal direction, respectively. Numerical illustrations with generalized Burgers–Fisher equation, Burgers–Huxley equation, and some physically relevant parabolic partial differential equations corroborate the theoretical analysis.

在本文中，我们描述了三次样条与高精度紧致公式相结合的方法，用于计算非线性对流占优扩散方程的近似解值，并将该方法应用于Burger型抛物型偏微分方程的几种变体。三次样条插值多项式与准变量网格序列的组合产生了两级隐式紧致计算方案，该方案落在提高的精度范围内。在考虑边界层现象对流占优的抛物线方程时，网格拉伸参数起着重要作用。与在统一网格网络上获得的结果相比，在准可变网格网络上进行离散化可以提供更准确的解决方案。利用紧凑的算子可以很容易地将所提出的方案扩展到奇异方程，从而保持顺序和精度。通过稳定性分析表明，该新方法是稳定的，在空间和时间方向上的精度分别为4级和2级。带有广义Burgers–Fisher方程，Burgers–Huxley方程以及一些与物理相关的抛物线偏微分方程的数值插图证实了理论分析。