Exact PDF equations and closure approximations for advective-reactive transport

D. Venturi, D. M. Tartakovsky, A. M. Tartakovsky, G. E. Karniadakis

Research output: Contribution to journalArticlepeer-review


Mathematical models of advection-reaction phenomena rely on advective flow velocity and (bio) chemical reaction rates that are notoriously random. By using functional integral methods, we derive exact evolution equations for the probability density function (PDF) of the state variables of the advection-reaction system in the presence of random transport velocity and random reaction rates with rather arbitrary distributions. These PDF equations are solved analytically for transport with deterministic flow velocity and a linear reaction rate represented mathematically by a heterogeneous and strongly-correlated random field. Our analytical solution is then used to investigate the accuracy and robustness of the recently proposed large-eddy diffusivity (LED) closure approximation [1]. We find that the solution to the LED-based PDF equation, which is exact for uncorrelated reaction rates, is accurate even in the presence of strong correlations and it provides an upper bound of predictive uncertainty.

Original languageEnglish (US)
Pages (from-to)323-343
Number of pages21
JournalJournal of Computational Physics
StatePublished - Jun 15 2013
Externally publishedYes


  • Geochemistry
  • Heterogeneous reaction
  • PDF methods
  • Stochastic modeling
  • Uncertainty quantification

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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