A fourth-order numerical method based on cubic B-spline functions has been proposed to solve a class of advection–diffusion equations. The proposed method has several advantageous features such as high accuracy and fast results with very small CPU time. We have applied the Crank–Nicolson method to solve the advection–diffusion equation. The stability analysis is performed, and the method is shown to be unconditionally stable. Error analysis is carried out to show that the proposed method has fourth-order convergence. The efficiency of the proposed B-spline method has been checked by applying on ten important advection–diffusion problems of three types, having Dirichlet, Neumann and periodic boundary conditions. Considered examples prove the mentioned advantages of the method. The computed results are also compared with those available in the literature, and it is found that our method is giving better results.

提出了一种基于三次B样条函数的四阶数值方法来求解一类对流扩散方程。所提出的方法具有几个有利的特征，例如高精度和快速结果，而CPU时间却非常短。我们已经应用了Crank–Nicolson方法来求解对流扩散方程。进行了稳定性分析，结果表明该方法是无条件稳定的。误差分析表明，该方法具有四阶收敛性。通过应用三种重要的对流扩散问题，分别具有Dirichlet，Neumann和周期边界条件这三种类型的十个重要对流扩散问题，检验了所提出的B样条方法的效率。实例证明了该方法的优点。