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

Abstract

In this paper, 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 $\theta$-method, $\theta\in[1/2, 1]$
($\theta=1$ corresponds to the back-ward Euler method and
$\theta=1/2$ corresponds to the Crank-Nicolson method) to discretize
in temporal direction and the quintic spline collocation method. The
convergence analysis of proposed method are discussed in detail, it
is justifying that the approximate solution converges to the exact
solution of orders $O(\Delta t + h^3)$ for the back-ward Euler
method and $O(\Delta t^2 + h^3)$ for the Crank–Nicolson method,
where k and h are mesh sizes in the time and space directions,
respectively. The proposed method is also shown to be
unconditionally stable. This scheme applied on some test examples,
the numerical results illustrate the efficiency of the method and
confirm the theoretical behavior of the rates of convergence.
Results shown by this method are found to be in good agreement with
the known exact solutions. The produced results are also seen to be
more accurate than some available results given in the literature.