# A quintic spline collocation method for solving time-dependent convection-diffusion problems

In: Tatra Mountains Mathematical Publications, vol. 80, no. 3
Abdelmajid El Hajaji - Abdelhafid Serghini - Said Melliani - Jalila El Ghordaf - Khalid Hilal

## Details:

Year, pages: 2021, 15 - 34
Language: eng
Keywords:
convection-diffusion equation, $\theta$-method, quintic, spline collocation method, convergence
Article type: Mathematics
Document type: scientific paper
In this paper, we develop a new numerical algorithm for solving a time dependent convection-diffusion equation with Dirichlet's type boundary conditions. The method comprises the horizontal method of lines for time integration and $θ$-method, $θ\in[1/2, 1]$ ($θ=1$ corresponds to the backward Euler method and $θ=1/2$ corresponds to the Crank-Nicolson method) to discretize in temporal direction and the quintic spline collocation method. The convergence analysis of proposed method is discussed in detail, and it justified that the approximate solution converges to the exact solution of orders $O(Δ t + h3)$ for the backward Euler method and $O(Δ t2 + h3)$ for the Crank-Nicolson method, where $Δ t$ and $h$ are mesh sizes in the time and space directions, respectively. It is also shown that the proposed method is unconditionally stable. This scheme is applied on some test examples, the numerical results illustrate the efficiency of the method and confirm the theoretical behaviour of the rates of convergence. Results shown by this method are in good agreement with the known exact solutions. The produced results are also more accurate than some available results given in the literature.