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Reduced order modelling of nonlinear cross-diffusion systems

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  • Karasözen, Bülent
  • Mülayim, Gülden
  • Uzunca, Murat
  • Yıldız, Süleyman

Abstract

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.

Suggested Citation

  • Karasözen, Bülent & Mülayim, Gülden & Uzunca, Murat & Yıldız, Süleyman, 2021. "Reduced order modelling of nonlinear cross-diffusion systems," Applied Mathematics and Computation, Elsevier, vol. 401(C).
    • Handle: RePEc:eee:apmaco:v:401:y:2021:i:c:s0096300321001065
    DOI: 10.1016/j.amc.2021.126058
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    References listed on IDEAS

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    1. Gambino, G. & Lombardo, M.C. & Sammartino, M., 2012. "Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross-diffusion," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(6), pages 1112-1132.
    2. Timo Reis & Tatjana Stykel, 2007. "Stability analysis and model order reduction of coupled systems," Mathematical and Computer Modelling of Dynamical Systems, Taylor & Francis Journals, vol. 13(5), pages 413-436, October.
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