Energy Stable Discontinuous Galerkin Finite Element Method for the Allen-Cahn Equation

Karasözen, Bülent
Sariaydin-Filibelioglu, Ayse
Yücel, 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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Sariaydin-Filibelioglu, Ayse; Karasözen, Bülent; Uzunca, Murat (2017-08-01)
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 ...
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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...
Time-Space Adaptive Method of Time Layers for the Advective Allen-Cahn Equation
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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.
Average Vector Field Splitting Method for Nonlinear Schrodinger Equation
Akkoyunlu, Canan; Karasözen, Bülent (2012-05-02)
The energy preserving average vector field integrator is applied to one and two dimensional Schrodinger equations with symmetric split-step method. The numerical results confirm the long-term preservation of the Hamiltonians, which is essential in simulating periodic waves.
Inverse problems for a semilinear heat equation with memory
Kaya, Müjdat; Çelebi, Okay; Department of Mathematics (2005)
In this thesis, we study the existence and uniqueness of the solutions of the inverse problems to identify the memory kernel k and the source term h, derived from First, we obtain the structural stability for k, when p=1 and the coefficient p, when g( )= . To identify the memory kernel, we find an operator equation after employing the half Fourier transformation. For the source term identification, we make use of the direct application of the final overdetermination conditions.
Citation Formats
B. Karasözen, A. Sariaydin-Filibelioglu, and H. Yücel, “Energy Stable Discontinuous Galerkin Finite Element Method for the Allen-Cahn Equation,” INTERNATIONAL JOURNAL OF COMPUTATIONAL METHODS, pp. 0–0, 2018, Accessed: 00, 2020. [Online]. Available: