Numerical method for advection diffusion equation using FEM and B-splines

In the present work, a comprehensive study of advection–diffusion equation is made using B-spline functions. Advection–diffusion equation has many physical applications such as dispersion of dissolved salts in groundwater, spread of pollutants in rivers and streams, water transfer, dispersion of tracers, and flow fast through porous media. Motivation behind the proposed scheme is to present a solution scheme which is easy to understand. Both linear and quadratic B-spline functions have been used in the present work to understand the basic aspect and advantages of the presented scheme. Along with this, some test examples are studied to observe the correctness of the numerical experiments. Finally, different comparisons are made to cross check the results obtained by the given scheme.

► This paper presents a detailed study on the advection diffusion equation that is related to turbulence theory and has similarity with the Navier–Stokes equation.
► Here an attempt has been taken to solve the above equation using linear and quadratic B-spline FEM.
► This method is suitable, if the unknown function is not enough smooth (i.e. not differentiable or differentiable only one time).
► The methods described play an important role in the literature for finding initial solutions.