On the energy preserving and efficient solutions of ginzburg-landau equation

In this thesis, numerical solution of one dimensional real Ginzburg-Landau equation has been considered by the use of discretization methods in both space and time. For the space discretization, spectral finite elemets method has been used, which results in a diagonal mass matrix. A Lyapunov energy functional is defined for real Ginzburg-Landau equation. Physically, the fundamental property of the so-called energy functional is that it decreases by the time progresses, and it becomes crucial to use a numerical scheme which preserves this energy decrease property in order to obtain reliable numerical approximations. In this thesis, for the time discretization, average vector field method has been used, which is an energy preserving, second order accurate method. Here, by energy preserving, it is meant the preservation of the energy decrease property of the Lyapunov energy functional. For the numerical solution of almost all the real life problems, it requires solution of linear systems of very large dimension. Since it takes a long time to solve, there has been a wide range of research concerning the fast solution of such large systems. In this thesis, reduced order modeling based on the proper orthogonal decomposition has been used for the fast numerical solution of real Ginzburg-Landau equation. The accuracy of numerical approximations and the high speed of the numerical scheme have been shown through test simulations.