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Title: Enforcing positivity in intrusive PC-UQ methods for reactive ODE systems

Abstract

We explore the relation between the development of a non-negligible probability of negative states and the instability of numerical integration of the intrusive Galerkin ordinary differential equation system describing uncertain chemical ignition. To prevent this instability without resorting to either multi-element local polynomial chaos (PC) methods or increasing the order of the PC representation in time, we propose a procedure aimed at modifying the amplitude of the PC modes to bring the probability of negative state values below a user-defined threshold. This modification can be effectively described as a filtering procedure of the spectral PC coefficients, which is applied on-the-fly during the numerical integration when the current value of the probability of negative states exceeds the prescribed threshold. We demonstrate the filtering procedure using a simple model of an ignition process in a batch reactor. This is carried out by comparing different observables and error measures as obtained by non-intrusive Monte Carlo and Gauss-quadrature integration and the filtered intrusive procedure. Lastly, the filtering procedure has been shown to effectively stabilize divergent intrusive solutions, and also to improve the accuracy of stable intrusive solutions which are close to the stability limits.

Authors:
 [1];  [2]
+ Show Author Affiliations
  1. Sandia National Lab. (SNL-CA), Livermore, CA (United States)
  2. Sapienza Univ. of Rome (Italy)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-CA), Livermore, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES)
OSTI Identifier:
1106517
Report Number(s):
SAND2013-6540J
Journal ID: ISSN 0021-9991; PII: S0021999114002484
Grant/Contract Number:  
AC04-94AL85000
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 270; Related Information: Proposed for publication in Journal of Computational Physics.; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Uncertainty quanti cation; polynomial chaos; intrusive; ordinary di erential equations; stability

Citation Formats

Najm, Habib N., and Valorani, Mauro. Enforcing positivity in intrusive PC-UQ methods for reactive ODE systems. United States: N. p., 2014. Web. doi:10.1016/j.jcp.2014.03.061.
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Najm, Habib N., & Valorani, Mauro. Enforcing positivity in intrusive PC-UQ methods for reactive ODE systems. United States. https://doi.org/10.1016/j.jcp.2014.03.061
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Najm, Habib N., and Valorani, Mauro. Sat . "Enforcing positivity in intrusive PC-UQ methods for reactive ODE systems". United States. https://doi.org/10.1016/j.jcp.2014.03.061. https://www.osti.gov/servlets/purl/1106517.
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@article{osti_1106517,
title = {Enforcing positivity in intrusive PC-UQ methods for reactive ODE systems},
author = {Najm, Habib N. and Valorani, Mauro},
abstractNote = {We explore the relation between the development of a non-negligible probability of negative states and the instability of numerical integration of the intrusive Galerkin ordinary differential equation system describing uncertain chemical ignition. To prevent this instability without resorting to either multi-element local polynomial chaos (PC) methods or increasing the order of the PC representation in time, we propose a procedure aimed at modifying the amplitude of the PC modes to bring the probability of negative state values below a user-defined threshold. This modification can be effectively described as a filtering procedure of the spectral PC coefficients, which is applied on-the-fly during the numerical integration when the current value of the probability of negative states exceeds the prescribed threshold. We demonstrate the filtering procedure using a simple model of an ignition process in a batch reactor. This is carried out by comparing different observables and error measures as obtained by non-intrusive Monte Carlo and Gauss-quadrature integration and the filtered intrusive procedure. Lastly, the filtering procedure has been shown to effectively stabilize divergent intrusive solutions, and also to improve the accuracy of stable intrusive solutions which are close to the stability limits.},
doi = {10.1016/j.jcp.2014.03.061},
journal = {Journal of Computational Physics},
number = ,
volume = 270,
place = {United States},
year = {Sat Apr 12 00:00:00 EDT 2014},
month = {Sat Apr 12 00:00:00 EDT 2014}
}
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https://doi.org/10.1016/j.jcp.2014.03.061
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