Simulations of nonlinear advection-diffusion models through various finite element techniques

In this study, the Burgers equation is analyzed both numerically and mathematically by considering various finite element based techniques including Galerkin, Taylor-Galerkin and collocation methods for spatial variation of the equation. The obtained time dependent ordinary differential equation system is approximately solved by ff-family of time approximation. All these methods are theoretically explained using cubic B-spline basis and weight functions for a strong form of the model equation. Von Neumann matrix stability analysis is performed for each of these methods and stability criteria are determined in terms of the problem parameters. Some challenging examples of the Burgers equation are numerically solved and compared with the literature and exact solutions. Also, the proposed techniques have been compared with each other in terms of their advantages and disadvantages depending on the problem types. The more advantageous method of the three, in comparison to the other two, has been found for the special cases of the present problem in detail.