POD Models for Positive Fields in Advection-Diffusion-Reaction Equations

Reduced-order models based on the proper orthogonal decomposition are important tools in modeling and control of nonlinear distributed parameter systems. However, the global support of the reduced-basis functions and basis orthogonality provide challenges for applications such as combustion and fire models. This is due to the fact that all basis functions, except the first, have regions with different signs. Thus it can be challenging to simultaneously constrain coefficients to preserve positivity while ensuring an accurate representation of the full-order model. In this paper we introduce a logarithmic transformation that allows us to maintain both features though introduces its own issues. We consider a simplified Arrhenius reaction model to illustrate the transformation and the bases that are obtained. Imposing a conservation constraint is easy when the bases are coupled, but the logarithmic transformation forces us to decouple the models. This method is compared to a standard POD-Galerkin model, which predicts negative densities.