Solving one-dimensional advection diffusion transport equation by using CDV wavelet basis

This work is concerned with the development of a scheme to obtain boundary conditions adapted representations of derivatives (differential operators) in the orthonormal wavelet bases of Daubechies family in $$\Omega = [0, \ 1] \subset {\mathbb{R}}$$ . The scheme is employed to find approximate solution of the (1+1)-D advection-diffusion-transport equation arising in the flow of contaminant fluid (water) through a porous medium (ground). Representation of derivatives adopting Dirichlet’s boundary conditions have been derived first. These representations are then used to reduce the aforesaid advection-diffusion-transport equation to a system of nonhomogeneous first order coupled ordinary differential equations. It is observed that representations of the derivatives derived here shift the non-homogeneous terms in the boundary conditions to non-homogeneous terms of the reduced ordinary differential equations. The resulting ordinary differential equations can be solved globally in time, in general, which helps to prevent the necessary analysis of time discretization such as estimate of step lengths, error etc. In case of convection dominated problems, the system of ordinary differential equations can be solved numerically with the aid of any efficient ODE solver. An estimate of $$L^2$$ -error in wavelet-Galerkin approximation of the unknown solution has been presented. A number of examples is given for numerical illustration. It is found that the scheme is efficient and user friendly.