Reduced Order Modelling for Reaction-Diffusion Equations with Cross Diffusion

Date

2017-04-29

Author

Mülayim, Gülden
Karasözen, Bülent
Küçükseyhan, Tuğba
Uzunca, Murat

Metadata

Show full item record

Item Usage Stats

169
views

0
downloads

In this work, we present reduced-order models (ROMs) for a nonlinear cross-diffusion problem from population dynamics, the Shigesada-Kawasaki-Teramoto (SKT) equation with Lotka-Volterra kinetics. The formation of the patterns of the SKT equation consists of a fast transient phase and a long stationary phase. Reduced order solutions are computed by separating the time into two time-intervals. In numerical experiments, we show for one- and two-dimensional SKT equations with pattern formation, the reduced-order solutions obtained in the time-windowed form, i.e., principal decomposition framework, are more accurate than the global proper orthogonal decomposition solutions obtained in the whole time interval. The finite-difference discretization of the SKT equation in space results in a system of linear-quadratic ordinary differential equations. The ROMs have the same linear-quadratic structure as the full order model. Using the linear-quadratic structure of the ROMs, the computation of the reduced-order solutions is further accelerated by the use of proper orthogonal decomposition in a tensorial framework so that the computations in the reduced system are independent of the full-order solutions. Furthermore, the prediction capabilities of the ROMs are illustrated for one- and two-dimensional patterns. Finally, we show that the entropy is decreasing by the reduced solutions, which is important for the global existence of solutions to the nonlinear cross-diffusion equations such as the SKT equation.

URI

https://hdl.handle.net/11511/76769

Conference Name

International Workshop On Mathematical Models İn Engineering, 2017

Collections

Graduate School of Applied Mathematics, Conference / Seminar

Suggestions

OpenMETU
Core

Reduced order modelling of nonlinear cross-diffusion systems Karasözen, Bülent; Uzunca, Murat; Yıldız, Süleyman (2021-07-15) In this work, we present reduced-order models (ROMs) for a nonlinear cross-diffusion problem from population dynamics, the Shigesada-Kawasaki-Teramoto (SKT) equation with Lotka-Volterra kinetics. The formation of the patterns of the SKT equation consists of a fast transient phase and a long stationary phase. Reduced order solutions are computed by separating the time into two time-intervals. In numerical experiments, we show for one- and two-dimensional SKT equations with pattern formation, the reduced-orde...
Bound states of a more general exponential screened Coulomb potential Ikhdair, Sameer M.; Sever, Ramazan (Springer Science and Business Media LLC, 2007-05-01) An alternative approximation scheme has been used in solving the Schrodinger equation to the more general case of exponential screened Coulomb potential, V(r) = -(a/r)[1 + (1 + br)e(-2br)]. The bound state energies of the 1s, 2s and 3s-states, together with the ground state wave function are obtained analytically upto the second perturbation term.
Bound states of the Dirac equation for the PT-symmetric generalized Hulthen potential by the Nikiforov-Uvarov method Egrifes, H; Sever, Ramazan (Elsevier BV, 2005-09-05) The one-dimensional Dirac equation is solved for the PT-symmetric generalized Hulthen potential. The Nikiforov-Uvarov method which is based on solving the second-order linear differential equations by reduction to a generalized equation of hypergeometric type is used to obtain exact energy eigenvalues and corresponding eigenfunctions.
Oscillation of nonlinear elliptic inequalities with p(x)-Laplacian ŞAHİNER, YETER; Zafer, Ağacık (2013-04-01) Sufficient conditions are obtained for the oscillation of solutions of half-linear and super-half-linear elliptic inequalities with p(x)-Laplacian. The results obtained are new even for the one-dimensional case.
Gardner's deformations of the Boussinesq equations Karasu, Atalay (IOP Publishing, 2006-09-15) Using the algebraic method of Gardner's deformations for completely integrable systems, we construct recurrence relations for densities of the Hamiltonians for the Boussinesq and the Kaup-Boussinesq equations. By extending the Magri schemes for these equations, we obtain new integrable systems adjoint with respect to the initial ones and describe their Hamiltonian structures and symmetry properties.

Citation Formats

G. Mülayim, B. Karasözen, T. Küçükseyhan, and M. Uzunca, “Reduced Order Modelling for Reaction-Diffusion Equations with Cross Diffusion,” presented at the International Workshop On Mathematical Models İn Engineering, 2017, Ankara, Türkiye, 2017, Accessed: 00, 2021. [Online]. Available: https://hdl.handle.net/11511/76769.