Comparison between results of solution of Burgers’ equation and Laplace’s equation by Galerkin and least-square finite element methods

• Published in: Applied water science Vol. 8; no. 1; pp. 1 - 10
• Main Authors: Adib, Arash, Poorveis, Davood, Mehraban, Farid
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2018; Springer Nature B.V
• Subjects: Aquatic Pollution; Burgers equation; Comparative Law; Deformation; Dye dispersion; Earth and Environmental Science; Earth dams; Earth Sciences; Finite element method; Galerkin method; Groundwater table; Hydrogeology; Industrial and Production Engineering; International & Foreign Law; Laplace equation; Least squares; Mathematical analysis; Mathematical models; Methods; Nanotechnology; Original Article; Pressure head; Private International Law; Seepage; Shock waves; Waste Water Technology; Water Industry/Water Technologies; Water Management; Water Pollution Control; Water table; Numerical methods; The Laplace’s equation; Hyperbolic equations; Earth dam; Elliptical equations; Shock wave; The Burgers’ equation
• ISSN: 2190-5487; 2190-5495
• DOI: 10.1007/s13201-018-0683-0

In this research, two equations are considered as examples of hyperbolic and elliptic equations. In addition, two finite element methods are applied for solving of these equations. The purpose of this research is the selection of suitable method for solving each of two equations. Burgers’ equation is a hyperbolic equation. This equation is a pure advection (without diffusion) equation. This equation is one-dimensional and unsteady. A sudden shock wave is introduced to the model. This wave moves without deformation. In addition, Laplace’s equation is an elliptical equation. This equation is steady and two-dimensional. The solution of Laplace’s equation in an earth dam is considered. By solution of Laplace’s equation, head pressure and the value of seepage in the directions X and Y are calculated in different points of earth dam. At the end, water table is shown in the earth dam. For Burgers’ equation, least-square method can show movement of wave with oscillation but Galerkin method can not show it correctly (the best method for solving of the Burgers’ equation is discrete space by least-square finite element method and discrete time by forward difference.). For Laplace’s equation, Galerkin and least square methods can show water table correctly in earth dam.