Abstract:
In this Thesis, new finite difference schemes are developed for solving one-dimensional advection-diffusion equation (ADE) with and without a source term. The developed schemes are based on the weighted finite difference method and higher order finite difference approximation. Firstly, by changing the values of weighted parameters, explicit forward in time centred in space (FTCS) and implicit backward in time centred in space (BTCS) schemes are derived and applied in both Dirichlet and Neumann boundary conditions. Discretization of ADEs resulted in a tri-diagonal system of linear algebraic equations which are solved by Thomas algorithm. For both FTCS and BTCS, the consistency and the stability of the schemes have been investigated. The Von-Neumann stability analysis is used to analyse the stabilities of the developed schemes. Convergence rates of the schemes are also determined numerically and it is found that they coincide with the theoretical result. Secondly, a new higher order finite difference scheme is derived using Taylor series expansion. In developing this scheme, a sixth order central difference in space and a first order forward difference in time were employed. The efficiency of the developed schemes was illustrated by solving different ADEs with both Dirichlet and Neumann boundary conditions. It is concluded that the developed schemes are in very good agreement with the analytical solutions.
