Discontinuous galerkin finite elements method with structure preserving time integrators for gradient flow equations

Sarıaydın Filibelioğlu, Ayşe
Gradient flows are energy driven evolutionary equations such that the energy decreases along solutions. There have been surprisingly a large number of well-known partial differential equations (PDEs) which have the structure of a gradient flow in different research areas such as fluid dynamics, image processing, biology and material sciences. In this study, we focus on two systems which can be modeled by gradient flows;Allen-Cahn and Cahn-Hilliard equations. These equations model the phase separation in material science. Since an essential feature of the Allen-Cahn and Cahn-Hilliard equations is the energy decreasing property, it is important to design efficient and accurate numerical schemes that satisfy the corresponding energy decreasing property. We have used symmetric interior penalty Galerkin (SIPG) method to discretize the Allen-Cahn and Cahn-Hilliard equations in space. The resulting large system of ordinary differential equations (ODEs) as a gradient system are solved by the energy stable (energy decreasing) time integrators: implicit Euler and average vector field (AVF) methods. We have shown that implicit Euler and AVF time integrators coupled with SIPG method are unconditionally energy stable. Numerical results for both equations with polynomial and logarithmic energy functions, and constant and variable mobility functions illustrate the efficiency and accuracy of this approach. Advective Allen-Cahn equation is the simplest model of surface tension in the droplet breakup phenomena. The small surface time scale and convective time scale lead to unphysical oscillations in the solution. In contrast to the discretization of Allen-Cahn and Cahn-Hilliard equations using the method of lines, the advective Allen-Cahn equation is first discretized in time using implicit Euler method and the resulting sequence of semi–linear elliptic equations are solved with an adaptive algorithm. This corresponds to Rothe’s method. As a remedy of unphysical oscillations, an adaptive version of SIPG method based on residual based a posteriori error estimate is applied. Numerical results for convection dominated Allen-Cahn equation show the performance of adaptive algorithm.


Discontinuous galerkin methods for time-dependent convection dominated optimal control problems
Akman, Tuğba; Karasözen, Bülent; Department of Scientific Computing (2011)
Distributed optimal control problems with transient convection dominated diffusion convection reaction equations are considered. The problem is discretized in space by using three types of discontinuous Galerkin (DG) method: symmetric interior penalty Galerkin (SIPG), nonsymmetric interior penalty Galerkin (NIPG), incomplete interior penalty Galerkin (IIPG). For time discretization, Crank-Nicolson and backward Euler methods are used. The discretize-then-optimize approach is used to obtain the finite dimensi...
Boundary element method solution of initial and boundary value problems in fluid dynamics and magnetohydrodynamics
Bozkaya, Canan; Tezer, Münevver; Department of Mathematics (2008)
In this thesis, the two-dimensional initial and boundary value problems invol\-ving convection and diffusion terms are solved using the boundary element method (BEM). The fundamental solution of steady magnetohydrodynamic (MHD) flow equations in the original coupled form which are convection-diffusion type is established in order to apply the BEM directly to these coupled equations with the most general form of wall conductivities. Thus, the solutions of MHD flow in rectangular ducts and in infinite regions...
Numerical simulation of advective Lotka-Volterra systems by discontinuous Galerkin method
Aktaş, Senem; Karasözen, Bülent; Uzunca, Murat; Department of Scientific Computing (2014)
In this thesis, we study numerically advection-diffusion-reaction equations arising from Lotka-Volterra models in river ecosystems characterized by unidirectional flow. We consider two and three species models which include competition, coexistence and extinction depending on the parameters. The one dimensional models are discretized by interior penalty discontinuous Galerkin model in space. For time discretization, fully implicit backward Euler method and semi-implicit IMEX-BDF methods are used. Numerical ...
Distributed Optimal Control Problems Governed by Coupled Convection Dominated PDEs with Control Constraints
Yücel, Hamdullah (2013-08-30)
We study the numerical solution of control constrained optimal control problems governed by a system of convection diffusion equations with nonlinear reaction terms, arising from chemical processes. Control constraints are handled by using the primal-dual active set algorithm as a semi-smooth Newton method or by adding a Moreau-Yosida-type penalty function to the cost functional. An adaptive mesh refinement indicated by a posteriori error estimates is applied for both approaches.
Energy preserving model order reduction of the nonlinear Schrodinger equation
Karasözen, Bülent (2018-12-01)
An energy preserving reduced order model is developed for two dimensional nonlinear Schrodinger equation (NLSE) with plane wave solutions and with an external potential. The NLSE is discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of Hamiltonian ordinary differential equations are integrated in time by the energy preserving average vector field (AVF) method. The mass and energy preserving reduced order model (ROM) is constructed by proper orth...
Citation Formats
A. Sarıaydın Filibelioğlu, “Discontinuous galerkin finite elements method with structure preserving time integrators for gradient flow equations,” Ph.D. - Doctoral Program, Middle East Technical University, 2015.