# 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. 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:

- Sandia National Laboratories, Livermore, CA 94551 (United States)
- Mechanical and Aerospace Engineering Dept., Sapienza University of Rome, Rome (Italy)

- Publication Date:

- OSTI Identifier:
- 22314884

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 270; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CHAOS THEORY; COMPARATIVE EVALUATIONS; DIFFERENTIAL EQUATIONS; GAUSS FUNCTION; MATHEMATICAL SOLUTIONS; MONTE CARLO METHOD; POLYNOMIALS; PROBABILITY; QUADRATURES; 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.

```
Najm, Habib N., & Valorani, Mauro.
```*Enforcing positivity in intrusive PC-UQ methods for reactive ODE systems*. United States. doi:10.1016/J.JCP.2014.03.061.

```
Najm, Habib N., and Valorani, Mauro. Fri .
"Enforcing positivity in intrusive PC-UQ methods for reactive ODE systems". United States.
doi:10.1016/J.JCP.2014.03.061.
```

```
@article{osti_22314884,
```

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. 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 = {Fri Aug 01 00:00:00 EDT 2014},

month = {Fri Aug 01 00:00:00 EDT 2014}

}