A Study of Some Finite Difference Schemes for a Unidirectional Stochastic Transport Equation

Abstract

We study a uni-directional hyperbolic transport equation, with a homogeneous stochastic transport velocity, solved by Monte Carlo simulation. Several nite di erence schemes are applied to the deterministic problem in each Monte Carlo iteration, and the numerical solution of the stochastic problem is compared to analytical solutions derived in the paper. We present both a theoretical analysis and summarized results from extensive numerical experiments. The behavior of the various schemes depend on the stochastic properties of the problem, and there are new demands to the schemes when they are used as part of a Monte Carlo simulation. For example, schemes that are very oscillatory for a single deterministic problem, like the leapfrog scheme, turn out to be e cient and accurate for the corresponding stochastic problem.

Uncertainty quantification for linear hyperbolic equations with stochastic process or random field coefficients

“…We derive a closed form expression for the moments of the distribution of the solution. We thereby extend the result found in [17] and [6]. Furthermore, we introduce a second order (in space and time) Monte Carlo method to approximate the solution.…”

“…which do not depend on space or time. For instance, the authors in [17,5,3] present both theoretical results and numerical approximations. In [6] the authors present expressions for the distribution of the solution of a linear advection equation with a time-dependent velocity, given in terms of the probability density function of the underlying integral of the stochastic process.…”

A note on the Riemann problem for the random transport equation

“…The first is based on the Monte Carlo method which, in general, demands massive numerical simulations (see [8], for example), and the second is based on effective equations (see [3], for example), deterministic differential equations whose solutions are the statistical means of (3).…”

Dynamically Orthogonal Numerical Schemes for Efficient Stochastic Advection and Lagrangian Transport

“…For model order reduction of SPDEs, classic methods such as polynomial chaos Downloaded 12/12/18 to 18.51.0.96. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php [56,28,84,13], proper orthogonal decomposition (POD) [26,60], dynamic mode decomposition (DMD) [61,66,78,82,31], and stochastic Galerkin schemes and adjoint methods [10,7] assume a priori choices of time-independent modes \bfitu i (\bfitx ) and/or rely on Gaussianity assumptions on the probability distribution of the coefficients \zeta i . For example, the popular data POD [26] and DMD [66] methods suggest extracting timeindependent modes \bfitu i (\bfitx ) that respectively best represent the variability (for the POD method) or the approximate linear dynamics (for the DMD method) of a series of snapshots \bfitu (t k , \bfitx , \omega 0 ) for a given observed or simulated realization \omega 0 .…”

“…How to adapt these schemes for reduced-order numerical advection, which cannot afford examining the realizations individually, is therefore particularly challenging [77,80,65]. This explains in part why many stochastic advection attempts have essentially restricted themselves to 1D applications [19,28,13,56] or simplified 2D cases that do not exhibit strong shocks [81].…”