Numerical solution of semi-linear advection-diffusion-reaction equations by discontinuous galerkin methods

Yıldız, Süleyman
IIn this thesis, we study splitting methods for semi-linear advection-diffusion-reaction (ADR) equations which are discretized by the symmetric interior penalty Galerkin (SIPG) method in space. For the time integration Rosenbrock methods are used with Strang splitting. The linear system of equations are solved iteratively by preconditioned generalized minimum residual method (GMRES). Numerical experiments for ADR equations with different type nonlinearities demonstrate the effectiveness of the proposed approach.


Time-Space Adaptive Method of Time Layers for the Advective Allen-Cahn Equation
UZUNCA, MURAT; Karasözen, Bülent; Sariaydin-Filibelioglu, Ayse (2015-09-18)
We develop an adaptive method of time layers with a linearly implicit Rosenbrock method as time integrator and symmetric interior penalty Galerkin method for space discretization for the advective Allen-Cahn equation with nondivergence-free velocity fields. Numerical simulations for convection dominated problems demonstrate the accuracy and efficiency of the adaptive algorithm for resolving the sharp layers occurring in interface problems with small surface tension.
Numerical solution of the Fisher’s equation with discontinous Galerkin method
Özsoy, Fehmi; Karasözen, Bülent; Department of Scientific Computing (2015)
In this thesis, the Fisher’s equation is discretized in space with the symmetric interior point discontinuous Galerkin (SDIPG). As time integrator Kahan’s method is used, which is n efficient linearly implicit time integrator for PDE with quadratic nonlinearities like the Fisher’s equation. Numerical results for the SIPG method, Kahan’s method and mid-point method confirm the theoretically predicted convergence orders in space and time. Travelling waves with steep fronts are numerically well resolved in rea...
Multigrid methods for optimal control problems governed by convection-diffusion equations
Arslantaş, Özgün Murat; Karasözen, Bülent; Yücel, Hamdullah; Department of Scientific Computing (2015)
Linear-quadratic optimal control problems governed by partial differential equations proved themselves important through their use in many real life applications. In order to solve the large scale linear system of equations that results from optimality conditions of the optimization problem, efficient solvers are required. For this purpose, multigrid methods, with an ordering technique to deal with the dominating convection, can be good candidates. This thesis investigates an application of the multigrid me...
Distributed Optimal Control of Diffusion Convection Reaction Equations Using Discontinuous Galerkin Methods
Yücel, Hamdullah; Karasözen, Bülent (2011-09-09)
We discuss the symmetric interior penalty Galerkin (SIPG) method, the nonsymmetric interior penalty Galerkin (NIPG) method, and the incomplete interior penalty Galerkin (IIPG) method for the discretization of optimal control problems governed by linear diffusion-convection-reaction equations. For the SIPG discretization the discretize-then-optimize (DO) and the optimize-then-discretize (OD) approach lead to the same discrete systems and in both approaches the observed L 2 convergence for states and controls...
Transient simulation of radiating flows
Selçuk, Nevin; Ayranci, I; Tarhan, T (2005-06-01)
Time-dependent Navier-Stokes equations are solved in conjunction with the radiative transfer equation by coupling a previously developed direct numerical simulation-based computational fluid dynamics code to an existing radiation code, both based on the method of lines approach. The temperature profiles predicted by the coupled code are validated against steady-state solutions available in the literature for laminar, axisymmetric, hydrodynamically developed flow of a gray, absorbing, emitting fluid in a hea...
Citation Formats
S. Yıldız, “Numerical solution of semi-linear advection-diffusion-reaction equations by discontinuous galerkin methods,” M.S. - Master of Science, Middle East Technical University, 2016.