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, 55:9 | Cite as

Nonlinear Galerkin methods for a system of PDEs with Turing instabilities

  • Konstantinos Spiliotis
  • Lucia Russo
  • Francesco Giannino
  • Salvatore Cuomo
  • Constantinos Siettos
  • Gerardo Toraldo
Article

Abstract

We address and discuss the application of nonlinear Galerkin methods for the model reduction and numerical solution of partial differential equations (PDE) with Turing instabilities in comparison with standard (linear) Galerkin methods. The model considered is a system of PDEs modelling the pattern formation in vegetation dynamics. In particular, by constructing the approximate inertial manifold on the basis of the spectral decomposition of the solution, we implement the so-called Euler–Galerkin method and we compare its efficiency and accuracy versus the linear Galerkin methods. We compare the efficiency of the methods by (a) the accuracy of the computed bifurcation points, and, (b) by the computation of the Hausdorff distance between the limit sets obtained by the Galerkin methods and the ones obtained with a reference finite difference scheme. The efficiency with respect to the required CPU time is also accessed. For our illustrations we used three different ODE time integrators, from the Matlab ODE suite. Our results indicate that the performance of the Euler–Galerkin method is superior compared to the linear Galerkin method when either explicit or linearly implicit time integration scheme are adopted. For the particular problem considered, we found that the dimension of approximate inertial manifold is strongly affected by the lenght of the spatial domain. Indeeed, we show that the number of modes required to accurately describe the long time Turing pattern forming solutions increases as the domain increases.

Keywords

Inertial manifold Reaction–diffusion system Pattern formation Dissipative PDE 

Mathematics Subject Classification

65N30 37L65 35B40 35B36 35B38 35B32 

Notes

Acknowledgements

K.S., F.G., S.C. were supported by the grant “Programma di finanziamento della ricerca di Ateneo 2015” of the University of Naples Federico II, Italy.

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Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  • Konstantinos Spiliotis
    • 1
  • Lucia Russo
    • 2
  • Francesco Giannino
    • 1
  • Salvatore Cuomo
    • 3
  • Constantinos Siettos
    • 4
  • Gerardo Toraldo
    • 3
  1. 1.Laboratory of Applied Ecology and System Dynamics, Department of Agricultural SciencesUniversity of Naples Federico IINaplesItaly
  2. 2.Istituto di Ricerche sulla CombustioneConsiglio Nazionale delle RicercheNaplesItaly
  3. 3.Department of Mathematics and Applications “Renato Caccioppoli”University of Naples Federico IINaplesItaly
  4. 4.School of Applied Mathematics and Physical SciencesNational Technical University of AthensAthensGreece

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