Reduced Order Modelling for Reaction-Diffusion Equations with Cross Diffusion

Date

2017-04-29

Author

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

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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

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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.