Cumulative distribution function solutions of advection–reaction equations with uncertain parameters

Abstract

We derive deterministic cumulative distribution function (CDF) equations that govern the evolution of CDFs of state variables whose dynamics are described by the first-order hyperbolic conservation laws with uncertain coefficients that parametrize the advective flux and reactive terms. The CDF equations are subjected to uniquely specified boundary conditions in the phase space, thus obviating one of the major challenges encountered by more commonly used probability density function equations. The computational burden of solving CDF equations is insensitive to the magnitude of the correlation lengths of random input parameters. This is in contrast to both Monte Carlo simulations (MCSs) and direct numerical algorithms, whose computational cost increases as correlation lengths of the input parameters decrease. The CDF equations are, however, not exact because they require a closure approximation. To verify the accuracy and robustness of the large-eddydiffusivity closure, we conduct a set of numerical experiments which compare the CDFs computed with the CDF equations with those obtained via MCSs. This comparison demonstrates that the CDF equations remain accurate over a wide range of statistical properties of the two input parameters, such as their correlation lengths and variance of the coefficient that parametrizes the advective flux.

The method of distributions for dispersive transport in porous media with uncertain hydraulic properties

“…Finally, localizing (A6), i.e., assuming that the CDF gradient rF C varies slowly in space and time and, hence, can be taken outside the integral, yields a dispersive closure Qðx; tÞ % 2D m rF C , where the macrodispersion tensor D m ðx; tÞ is defined by (5). This expression for Q is similar to the large-eddy-diffusivity closure developed by Boso et al [2014] for advection-reaction equations.…”

“…Derivation of CDF equations for advection‐dominated transport does not require an IEM closure and is often exact [ Boso et al ., ; Tartakovsky and Gremaud , ]. It is therefore reasonable to expect that the performance of the IEM‐based CDF equation depends on the Péclet number .…”

“…This bimodality is largely absent in the dispersion‐dominated regime ( ), resulting in significantly reduced approximation errors. For advection‐dominated regimes it is advisable to resort to a different closure of the CDF equation [ Boso et al ., ], with closure coefficients that account for random advection.…”

“…PDF equations originated in the statistical theory of turbulence [ Lundgren , ] and have since been modified to deal with uncertain flow velocity and/or reaction rate constants in advection‐reaction transport problems [ Lichtner and Tartakovsky , ; Shvidler and Karasaki , ; Tartakovsky et al ., ; Venturi et al ., ]. Like their PDF counterparts, CDF equations treat nonlinear reaction terms in the transport equations exactly, without resorting to linearization required by MDEs [ Boso et al ., ; Tartakovsky and Gremaud , ]. Other advantages of CDF equations include the uniqueness of boundary conditions in the event space, e.g., for any and t , and the numerical efficiency of having to find a solution that is a smooth function in c , which increases from 0 and 1.…”

Probabilistic Forecasting of Nitrogen Dynamics in Hyporheic Zone

“…The method of distributions (Tartakovsky & Gremaud, 2016) is another alternative to MCS, which overcomes this limitation of MDEs by deriving a deterministic differential equation for either PDF or CDF of solute concentration, rather than its first two moments. The PDF/CDF equations are exact for advection-reaction single-species transport in deterministic velocity fields regardless of the linear or nonlinear nature of the reaction term (Boso et al, 2014;Lichtner & Tartakovsky, 2003;Venturi et al, 2013). The presence of diffusion and/or dispersion requires a closure approximation (Boso & Tartakovsky, 2016).…”