  Dynamic Mode Decomposition (DMD) is an equation-free method that aims at
reconstructing the best linear fit from temporal datasets. In this paper, we
show that DMD does not provide accurate approximation for datasets describing
oscillatory dynamics, like spiral waves and relaxation oscillations, or
spatio-temporal Turing instability. Inspired from the classical "divide and
conquer" approach, we propose a piecewise version of DMD (pDMD) to overcome
this problem. The main idea is to split the original dataset in N submatrices
and then apply the exact (randomized) DMD method in each subset of the obtained
partition. We describe the pDMD algorithm in detail and we introduce some error
indicators to evaluate its performance when N is increased. Numerical
experiments show that very accurate reconstructions are obtained by pDMD for
datasets arising from time snapshots of some reaction-diffusion PDE systems,
like the FitzHugh-Nagumo model, the lambda-omega system and the DIB
morpho-chemical system for battery modeling.
