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Adaptive POD-DEIM correction for Turing pattern approximation in reaction–diffusion PDE systems

  • Alessandro Alla EMAIL logo , Angela Monti and Ivonne Sgura

Abstract

We investigate a suitable application of Model Order Reduction (MOR) techniques for the numerical approximation of Turing patterns, that are stationary solutions of reaction–diffusion PDE (RD-PDE) systems. We show that solutions of surrogate models built by classical Proper Orthogonal Decomposition (POD) exhibit an unstable error behaviour over the dimension of the reduced space. To overcome this drawback, first of all, we propose a POD-DEIM technique with a correction term that includes missing information in the reduced models. To improve the computational efficiency, we propose an adaptive version of this algorithm in time that accounts for the peculiar dynamics of the RD-PDE in presence of Turing instability. We show the effectiveness of the proposed methods in terms of accuracy and computational cost for a selection of RD systems, i.e., FitzHugh–Nagumo, Schnakenberg and the morphochemical DIB models, with increasing degree of nonlinearity and more structured patterns.

JEL Classification: 65M06; 35K57; 65F99; 65M22
  1. Code availability

    The MATLAB source code of the implementations used to compute the presented results can be downloaded from https://github.com/alessandroalla/PODcorrection.

Acknowledgment

A.A., A.M., and I.S. are members of the INdAM-GNCS activity group. The work of I.S. is supported by the MIUR (Italian Ministry of University and Research) through the project PRIN 2020, ‘Mathematics for Industry 4.0’, project No. 2020F3NCPX.

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Received: 2022-03-11
Revised: 2022-12-06
Accepted: 2023-01-05
Published Online: 2023-01-20
Published in Print: 2023-09-07

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