Ayşe Sarıaydın Filibelioğlu

Murat Uzunca, Bülent Karasözen, Ayşe Sarıaydın Filibelioğlu

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 non-divergence-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.

Bülent Karasözen, Ayşe Sarıaydın Filibelioğlu, Murat Uzunca

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 CH equation. Numerical results confirm the theoretical convergence rates and the performance of the proposed approach.

Bülent Karasözen, Ayşe Sarıaydın Filibelioğlu, Murat Uzunca

Allen--Cahn equation with constant and degenerate mobility, and with polynomial and logarithmic energy functionals is discretized using symmetric interior penalty discontinuous Galerkin (SIPG) finite elements in space. We show that the energy stable average vector field (AVF) method as the time integrator for gradient systems like the Allen-Cahn equation satisfies the energy decreasing property for the fully discrete scheme. The numerical results for one and two dimensional Allen-Cahn equation with periodic boundary condition, using adaptive time stepping, reveal that the discrete energy decreases monotonically, the phase separation and metastability phenomena can be observed and the ripening time is detected correctly.