The numerical study of advection–diffusion equations by the fourth-order cubic B-spline collocation method

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.