There is plenty of differential equations such as drift-diffusion ones that have a non-negative solution (positive initial and boundary conditions, positive forcing term). However the pod-based model reduction cannot assure that the positiveness is preserved. I would really like to know whether there is any derived model reduction approach that can guarantee the positiveness.
You may wish to consider using the balanced model reduction, which combines POD and the balanced realization theory. Details are in the following paper.
K. Willcox and J. Peraire. "Balanced Model Reduction via the Proper Orthogonal Decomposition", AIAA Journal, Vol. 40, No. 11 (2002), pp. 2323-2330.
The Willcox/Peraire variant of POD does not preserve positivity, either. If you talk about a time-dependent problem, then you need a positivity preserving discretization scheme for the spatial discretization. I am not an expert in this. But suppose you have such a scheme, then there is a variant of balanced trucnation that indeed does preserve positivity, see
Positivity preserving balanced truncation for descriptor systems
T Reis, E Virnik
SIAM Journal on Control and Optimization 48 (4), 2600-2619
A pdf is available on ResearchGate:
https://www.researchgate.net/profile/Elena_Virnik/publication/220259131_Positivity_Preserving_Balanced_Truncation_for_Descriptor_Systems/links/00b7d5174f1aa6ed84000000.pdf
Article Positivity Preserving Balanced Truncation for Descriptor Systems
Thanks, in fact the scheme i have used is positive but, as you stated, the projection to a lower dimensional space may not preserve this property.
Thank you very much.
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