Moving mesh discontinuous Galerkin methods for PDEs with traveling waves

Karasözen, Bülent
Kucukseyhan, T.
In this paper, a moving mesh discontinuous Galerkin (dG) method is developed for nonlinear partial differential equations (PDEs) with traveling wave solutions. The moving mesh strategy for one dimensional PDEs is based on the rezoning approach which decouples the solution of the PDE from the moving mesh equation. We show that the dG moving mesh method is able to resolve sharp wave fronts and wave speeds accurately for the optimal, arc-length and curvature monitor functions. Numerical results reveal the efficiency of the proposed moving mesh dG method for solving Burgers', Burgers'-Fisher and Schlogl (Nagumo) equations.


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.
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.
Energy preserving methods for lattice equations
Erdem, Özge; Karasözen, Bülent (2010-11-27)
Integral preserving methods, like the averaged vector field, discrete gradient and trapezoidal methods are to Poisson systems. Numerical experiments on the Volterra equations and integrable discretization of the nonlinear Schrodinger equation are presented.
A two-dimensional vibrational system with a strong nonlinear coupling is studied using a quantum-classical mixed mode self-consistent-field approach. The classical equations of motion as well as the time-dependent Schrodinger equation are solved for respective modes under the influence of the average fields generated by the other modes. This vibrational system was previously shown to be chaotic under classical mechanical treatment but quantum mechanical observations pointed out to highly regular behaviour...
Optimal control of convective FitzHugh-Nagumo equation
Uzunca, Murat; Kucukseyhan, Tugba; Yücel, Hamdullah; Karasözen, Bülent (2017-05-01)
We investigate smooth and sparse optimal control problems for convective FitzHugh Nagumo equation with traveling wave solutions in moving excitable media. The cost function includes distributed space time and terminal observations or targets. The state and adjoint equations are discretized in space by symmetric interior point Galerkin (SIPG) method and by backward Euler method in time. Several numerical results are presented for the control of the traveling waves. We also show numerically the validity of th...
Citation Formats
M. UZUNCA, B. Karasözen, and T. Kucukseyhan, “Moving mesh discontinuous Galerkin methods for PDEs with traveling waves,” APPLIED MATHEMATICS AND COMPUTATION, pp. 9–18, 2017, Accessed: 00, 2020. [Online]. Available: