Browsing by Author "Uzunca, Murat"

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    Energy Stable Discontinuous Galerkin Finite Element Method for the Allen-Cahn Equation
    (World Scientific Publ Co Pte Ltd, 2018) Karasozen, Bulent; Uzunca, Murat; Sariaydin-Filibelioglu, Ayse; Yucel, Hamdullah
    In this paper, we investigate numerical solution of Allen-Cahn equation with constant and degenerate mobility, and with polynomial and logarithmic energy functionals. We discretize the model equation by symmetric interior penalty Galerkin (SIPG) method in space, and by average vector field (AVF) method in time. We show that the energy stable AVF method as the time integrator for gradient systems like the Allen-Cahn equation satisfies the energy decreasing property for fully discrete scheme. Numerical results reveal that the discrete energy decreases monotonically, the phase separation and metastability phenomena can be observed, and the ripening time is detected correctly.
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    Article
    Energy Stable Interior Penalty Discontinuous Galerkin Finite Element Method for Cahn-Hilliard Equation
    (Walter de Gruyter Gmbh, 2017) Sariaydin-Filibelioglu, Ayse; Karasozen, Bulent; Uzunca, Murat
    An energy stable conservative method is developed for the Cahn-Hilliard (CH) equation with the degenerate mobility. The CH equation is discretized in space with the mass conserving symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting semi-discrete nonlinear system of ordinary differential equations are solved in time by the unconditionally energy stable average vector field (AVF) method. We prove that the AVF method preserves the energy decreasing property of the fully discretized CH equation. Numerical results for the quartic double-well and the logarithmic potential functions with constant and degenerate mobility confirm the theoretical convergence rates, accuracy and the performance of the proposed approach.