Simplified CSP analysis of a stiff stochastic ODE system

Maher Salloum, Alen Alexanderian, Olivier P. Le Maître, Habib N. Najm, Omar M. Knio*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

15 Scopus citations


We develop a simplified computational singular perturbation (CSP) analysis of a stochastic dynamical system. We focus on the case of parametric uncertainty, and rely on polynomial chaos (PC) representations to quantify its impact. We restrict our attention to a system that exhibits distinct timescales, and that tends to a deterministic steady state irrespective of the random inputs. A detailed analysis of eigenvalues and eigenvectors of the stochastic system Jacobian is conducted, which provides a relationship between the PC representation of the stochastic Jacobian and the Jacobian of the Galerkin form of the stochastic system. The analysis is then used to guide the application of a simplified CSP formalism that is based on relating the slow and fast manifolds of the uncertain system to those of a nominal deterministic system. Two approaches are specifically developed with the resulting simplified CSP framework. The first uses the stochastic eigenvectors of the uncertain system as CSP vectors, whereas the second uses the eigenvectors of the nominal system as CSP vectors. Numerical experiments are conducted to demonstrate the results of the stochastic eigenvalue and eigenvector analysis, and illustrate the effectiveness of the simplified CSP algorithms in addressing the stiffness of the system dynamics.

Original languageEnglish (US)
Pages (from-to)121-138
Number of pages18
JournalComputer Methods in Applied Mechanics and Engineering
StatePublished - Apr 1 2012
Externally publishedYes


  • CSP
  • Polynomial chaos
  • Random eigenvalues
  • Stiff system
  • Uncertain ODE

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • General Physics and Astronomy
  • Computer Science Applications


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