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

TL;DR

This paper introduces an adaptive POD-DEIM correction method to improve the accuracy and efficiency of reduced-order models for approximating Turing patterns in reaction-diffusion PDE systems, addressing instability issues of classical approaches.

Contribution

The paper proposes a novel adaptive POD-DEIM correction technique tailored for reaction-diffusion systems with Turing patterns, enhancing model stability and computational efficiency.

Findings

The adaptive correction improves accuracy over classical POD-DEIM.

The method reduces computational cost for complex RD systems.

Effective for models like FitzHugh-Nagumo, Schnackenberg, and DIB.

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, Schnackenberg and the morphochemical DIB models, with increasing degree of nonlinearity and more structured patterns.