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.


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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.
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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...
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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.