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

Abstract

In this study, the Burgers equation is analyzed both numerically and mathematically by considering various nite element based techniques including Galerkin, Taylor-Galerkin and collocation methods for spatial variation of the equation. The obtained time dependent ordinary di erential equation system is approximately solved by -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.

An implicit-explicit local method for parabolic partial differential equations

“…The exact solution of equation (41) under the consideration of cases (42)–(43) given in (Tunc and Sari, 2020) iswith Fourier coefficients…”

“…In Table 2, the fourth and sixth order IELDTM-ChCM algorithm has been compared with the FEM (Tunc and Sari, 2020), the FDM (Mukundan and Awasthi, 2015) and the exact solution for solving nonlinear advection-diffusion equation (41). As seen in Table 2, the proposed sixth-order method offers more accurate results using 20 times less dof than both finite element and FDMs.…”