Dynamically Orthogonal Research Articles

Dynamically orthogonal narrow-angle parabolic equations for stochastic underwater sound propagation. Part II: Applications.

The stochastic dynamically orthogonal (DO) narrow-angle parabolic equations (NAPEs) are exemplified and their properties and capabilities are described using three new two-dimensional stochastic range-independent and range-dependent test cases with uncertain sound speed field, bathymetry, and source location. We validate results against ground-truth deterministic analytical solutions and direct Monte Carlo (MC) predictions of acoustic pressure and transmission loss fields. We verify the stochastic convergence and computational advantages of the DO-NAPEs and discuss the differences with normal mode approaches. Results show that a single DO-NAPE simulation can accurately predict stochastic range-dependent acoustic fields and their non-Gaussian probability distributions, with computational savings of several orders of magnitude when compared to direct MC methods. With their coupling properties and their adaptation in range to the dominant uncertainties, the DO-NAPEs are shown to predict accurate statistics, from mean and variance to multiple modes and full probability distributions, and to provide excellent reconstructed realizations, from amplitudes and phases to other specific properties of complex realization fields.

Dynamically orthogonal narrow-angle parabolic equations for stochastic underwater sound propagation. Part I: Theory and schemes.

Robust informative acoustic predictions require precise knowledge of ocean physics, bathymetry, seabed, and acoustic parameters. However, in realistic applications, this information is uncertain due to sparse and heterogeneous measurements and complex ocean physics. Efficient techniques are thus needed to quantify these uncertainties and predict the stochastic acoustic wave fields. In this work, we derive and implement new stochastic differential equations that predict the acoustic pressure fields and their probability distributions. We start from the stochastic acoustic parabolic equation (PE) and employ the instantaneously-optimal Dynamically Orthogonal (DO) equations theory. We derive stochastic DO-PEs that dynamically reduce and march the dominant multi-dimensional uncertainties respecting the nonlinear governing equations and non-Gaussian statistics. We develop the dynamical reduced-order DO-PEs theory for the Narrow-Angle parabolic equation and implement numerical schemes for discretizing and integrating the stochastic acoustic fields.

A Study of The Stochastic Burgers’ Equation Using The Dynamical Orthogonal Method

In the current work, the stochastic Burgers’ equation is studied using the Dynamically Orthogonal (DO) method. The DO presents a low-dimensional representation for the stochastic fields. Unlike many other methods, it has a time-dependent property on both the spatial basis and stochastic coefficients, with flexible representation especially in the strongly transient and nonstationary problems. We consider a computational study and application of the DO method and compare it with the Polynomial Chaos (PC) method. For comparison, both the stochastic viscous and inviscid Burgers’ equations are considered. A hybrid approach, combining the DO and PC is proposed in case of deterministic initial conditions to overcome the singularities that occur in the DO method. The results are verified with the stochastic collocation method. Overall, we observe that the DO method has a higher rate of convergence as the number of modes increases. The DO method is found to be more efficient than PC for the same level of accuracy, especially for the case of high-dimensional parametric spaces. The inviscid Burgers’ equation is analyzed to study the shock wave formation when using the DO after suitable handling of the convective term. The results show that the sinusoidal wave shape is distorted and sharpened as the time evolves till the shock wave occurs.

High frequency stochastic acoustic wavefront propagation and joint ocean-acoustic inference: The GMM-DO wavefront

In marine applications, the value of accurate modeling and learning for stochastic acoustic propagation in uncertain ocean environments cannot be overstated. In this work, we derive stochastic theory and schemes for (i) modeling of high frequency acoustic propagation in uncertain ocean environments and (ii) joint Bayesian assimilation of ocean-acoustic measurements to infer fields, parameters, and uncertain model functions. We first obtain the Dynamically Orthogonal (DO) wavefront equations to solve for the stochastic extension of the Liouville Equation that governs the dynamics of acoustic wavefront in an augmented phase space. These DO wavefront equations provide the prior for the Gaussian Mixture Model—DO (GMM-DO) filter that completes joint physics-acoustics Bayesian inference using sparse observations. Specifically, given a set of receivers, the Eulerian nature of the DO wavefront equations allows for the efficient extraction of arrival time prior probability distributions. The GMM-DO Waverfront filter then combines these joint priors with arrival time measurements using Bayes rule, jointly inferring environmental properties (e.g., unknown source location and/or sound speed field), the acoustic wavefront distribution, and the arrival time distribution itself. We evaluate results using high-frequency applications, illustrating the estimation of mean fields and properties, but also of probability density distributions and model parameterizations.

Stochastic sound propagation using dynamically orthogonal parabolic equations

Accurate modeling of underwater acoustic propagation is challenging due to the complex ocean physics and acoustic dynamics and the need for resolving the wavelength of the propagating acoustic wave over large distances. These challenges are further amplified by the incomplete knowledge of the ocean environment and the acoustic parameters. These complexities thus lead to many sources of uncertainties in the governing models. In this work, we use our stochastic Dynamically Orthogonal (DO) framework to represent these uncertainties probabilistically in the acoustic Parabolic Equation (PE). These equations optimally represent the dominant uncertainties in the sound speed, density, bathymetry, and acoustic pressure fields. Starting from the governing PE, we derive range-evolution DO differential equations for the mean field, stochastic modes, and coefficients, hence preserving the nonlinearities and capturing the non-Gaussian statistics. The DO equations are implemented for the narrow-angle PE and higher-order Padé wide-angle PEs and are applied in range-dependent canonical test cases and realistic ocean environments with uncertain source location, source frequency, sound speed, and/or bathymetry fields. We highlight the computational advantages of our framework by comparing it to Monte Carlo predictions and show convergence of the probability density functions as the number of samples and/or modes is increased.

Bayesian learning of stochastic dynamical models

A new methodology for rigorous Bayesian learning of high-dimensional stochastic dynamical models is developed. The methodology performs parallelized computation of marginal likelihoods for multiple candidate models, integrating over all state variable and parameter values, and enabling a principled Bayesian update of model distributions. This is accomplished by leveraging the dynamically orthogonal (DO) evolution equations for uncertainty prediction in a dynamic stochastic subspace and the Gaussian Mixture Model-DO filter for inference of nonlinear state variables and parameters, using reduced-dimension state augmentation to accommodate models featuring uncertain parameters. Overall, the joint Bayesian inference of the state, model equations, geometry, boundary conditions, and initial conditions is performed. Results are exemplified using two high-dimensional, nonlinear simulated fluid and ocean systems. For the first, limited measurements of fluid flow downstream of an obstacle are used to perform joint inference of the obstacle’s shape, the Reynolds number, and the O(105) fluid velocity state variables. For the second, limited measurements of the concentration of a microorganism advected by an uncertain flow are used to perform joint inference of the microorganism’s reaction equation and the O(105) microorganism concentration and ocean velocity state variables. When the observations are sufficiently informative about the learning objectives, we find that our posterior model probabilities correctly identify either the true model or the most plausible models, even in cases where a human would be challenged to do the same.

Real-time reduced-order modeling of stochastic partial differential equations via time-dependent subspaces

We present a new methodology for the real-time reduced-order modeling of stochastic partial differential equations called the dynamically/bi-orthonormal (DBO) decomposition. In this method, the stochastic fields are approximated by a low-rank decomposition to spatial and stochastic subspaces. Each of these subspaces is represented by a set of orthonormal time-dependent modes. We derive exact evolution equations of these time-dependent modes and the evolution of the factorization of the reduced covariance matrix. We show that DBO is equivalent to the dynamically orthogonal (DO) [1] and bi-orthogonal (BO) [2] decompositions via linear and invertible transformation matrices that connect DBO to DO and BO. However, DBO shows several improvements compared to DO and BO: (i) DBO performs better than DO and BO for cases with ill-conditioned covariance matrix; (ii) In contrast to BO, the issue of eigenvalue crossing is not present in the DBO formulation; (iii) In contrast to DO, the stochastic modes are orthonormal, which leads to more accurate representation of the stochastic subspace. We study the convergence properties of the method and compare it to the DO and BO methods. For demonstration, we consider three cases: (i) stochastic linear advection equation, (ii) stochastic Burgers' equation, and (iii) stochastic incompressible flow over a bump in a channel. Overall we observe improvements in the numerical accuracy of DBO compared against DO and BO.

Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks

One of the open problems in scientific computing is the long-time integration of nonlinear stochastic partial differential equations (SPDEs). We address this problem by taking advantage of recent advances in scientific machine learning and the dynamically orthogonal (DO) and bi-orthogonal (BO) methods for representing stochastic processes. Specifically, we propose two new Physics-Informed Neural Networks (PINNs) for solving time-dependent SPDEs, namely the NN-DO/BO methods, which incorporate the DO/BO constraints into the loss function with an implicit form instead of generating explicit expressions for the temporal derivatives of the DO/BO modes. Hence, the proposed methods overcome some of the drawbacks of the original DO/BO methods: we do not need the assumption that the covariance matrix of the random coefficients is invertible as in the original DO method, and we can remove the assumption of no eigenvalue crossing as in the original BO method. Moreover, the NN-DO/BO methods can be used to solve time-dependent stochastic inverse problems with the same formulation and computational complexity as for forward problems. We demonstrate the capability of the proposed methods via several numerical examples: (1) A linear stochastic advection equation with deterministic initial condition where the original DO/BO method would fail; (2) Long-time integration of the stochastic Burgers' equation with many eigenvalue crossings during the whole time evolution where the original BO method fails. (3) Nonlinear reaction diffusion equation: we consider both the forward and the inverse problem, including noisy initial data, to investigate the flexibility of the NN-DO/BO methods in handling inverse and mixed type problems. Taken together, these simulation results demonstrate that the NN-DO/BO methods can be employed to effectively quantify uncertainty propagation in a wide range of physical problems.

A Geometric Approach to Dynamical Model Order Reduction

Any model order reduced dynamical system that evolves a modal decomposition to approximate the discretized solution of a stochastic PDE can be related to a vector field tangent to the manifold of fixed rank matrices. The dynamically orthogonal (DO) approximation is the canonical reduced-order model for which the corresponding vector field is the orthogonal projection of the original system dynamics onto the tangent spaces of this manifold. The embedded geometry of the fixed rank matrix manifold is thoroughly analyzed. The curvature of the manifold is characterized and related to the smallest singular value through the study of the Weingarten map. Differentiability results for the orthogonal projection onto embedded manifolds are reviewed and used to derive an explicit dynamical system for tracking the truncated singular value decomposition (SVD) of a time-dependent matrix. It is demonstrated that the error made by the DO approximation remains controlled under the minimal condition that the original solution stays close to the low rank manifold, which translates into an explicit dependence of this error on the gap between singular values. The DO approximation is also justified as the dynamical system that applies instantaneously the SVD truncation to optimally constrain the rank of the reduced solution. Riemannian matrix optimization is investigated in this extrinsic framework to provide algorithms that adaptively update the best low rank approximation of a smoothly varying matrix. The related gradient flow provides a dynamical system that converges to the truncated SVD of an input matrix for almost every initial datum.

Model Order Reduction for Stochastic Dynamical Systems with Continuous Symmetries

Stochastic dynamical systems with continuous symmetries arise commonly in nature and often give rise to coherent spatio-temporal patterns. However, because of their random locations, these patterns are not captured well by current order reduction techniques, and a large number of modes is typically necessary for an accurate solution. In this work, we introduce a new methodology for efficient order reduction of such systems by combining (i) the method of slices [C. W. Rowley and J. E. Marsden, Phys. D, 142 (2000), pp. 1--19; S. Froehlich and P. Cvitanović, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), pp. 2074--2084], a symmetry reduction tool, and (ii) any standard order reduction technique, resulting in efficient mixed symmetry-dimensionality reduction schemes. In particular, using the dynamically orthogonal (DO) equations [T. P. Sapsis and P. F. J. Lermusiaux, Phys. D, 238 (2009), pp. 2347--2360] in the second step, we obtain a novel nonlinear symmetry-reduced dynamically orthogonal (SDO) scheme. We demonstrate the performance of the SDO scheme on stochastic solutions of the one-dimensional Korteweg--de Vries and two-dimensional Navier--Stokes equations.

A Gaussian Mixture Model Smoother for Continuous Nonlinear Stochastic Dynamical Systems: Theory and Scheme

Retrospective inference through Bayesian smoothing is indispensable in geophysics, with crucial applications in ocean and numerical weather estimation, climate dynamics, and Earth system modeling. However, dealing with the high-dimensionality and nonlinearity of geophysical processes remains a major challenge in the development of Bayesian smoothers. Addressing this issue, a novel subspace smoothing methodology for high-dimensional stochastic fields governed by general nonlinear dynamics is obtained. Building on recent Bayesian filters and classic Kalman smoothers, the fundamental equations and forward–backward algorithms of new Gaussian Mixture Model (GMM) smoothers are derived, for both the full state space and dynamic subspace. For the latter, the stochastic Dynamically Orthogonal (DO) field equations and their time-evolving stochastic subspace are employed to predict the prior subspace probabilities. Bayesian inference, both forward and backward in time, is then analytically carried out in the dominant stochastic subspace, after fitting semiparametric GMMs to joint subspace realizations. The theoretical properties, varied forms, and computational costs of the new GMM smoother equations are presented and discussed.

A robust bi-orthogonal/dynamically-orthogonal method using the covariance pseudo-inverse with application to stochastic flow problems

We develop a new robust methodology for the stochastic Navier–Stokes equations based on the dynamically-orthogonal (DO) and bi-orthogonal (BO) methods [1–3]. Both approaches are variants of a generalized Karhunen–Loève (KL) expansion in which both the stochastic coefficients and the spatial basis evolve according to system dynamics, hence, capturing the low-dimensional structure of the solution. The DO and BO formulations are mathematically equivalent [3], but they exhibit computationally complimentary properties. Specifically, the BO formulation may fail due to crossing of the eigenvalues of the covariance matrix, while both BO and DO become unstable when there is a high condition number of the covariance matrix or zero eigenvalues. To this end, we combine the two methods into a robust hybrid framework and in addition we employ a pseudo-inverse technique to invert the covariance matrix. The robustness of the proposed method stems from addressing the following issues in the DO/BO formulation: (i) eigenvalue crossing: we resolve the issue of eigenvalue crossing in the BO formulation by switching to the DO near eigenvalue crossing using the equivalence theorem and switching back to BO when the distance between eigenvalues is larger than a threshold value; (ii) ill-conditioned covariance matrix: we utilize a pseudo-inverse strategy to invert the covariance matrix; (iii) adaptivity: we utilize an adaptive strategy to add/remove modes to resolve the covariance matrix up to a threshold value. In particular, we introduce a soft-threshold criterion to allow the system to adapt to the newly added/removed mode and therefore avoid repetitive and unnecessary mode addition/removal. When the total variance approaches zero, we show that the DO/BO formulation becomes equivalent to the evolution equation of the Optimally Time-Dependent modes [4]. We demonstrate the capability of the proposed methodology with several numerical examples, namely (i) stochastic Burgers equation: we analyze the performance of the method in the presence of eigenvalue crossing and zero eigenvalues; (ii) stochastic Kovasznay flow: we examine the method in the presence of a singular covariance matrix; and (iii) we examine the adaptivity of the method for an incompressible flow over a cylinder where for large stochastic forcing thirteen DO/BO modes are active.

Dynamically orthogonal equations for stochastic underwater sound propagation

Grand challenges in ocean acoustic propagation and inference are to accurately capture the dynamic environmental uncertainties and to predict the evolving probability density distribution of stochastic acoustic waves, all efficiently and rigorously, using the governing partial differential equations (PDEs). To start addressing these needs, the stochastic dynamically orthogonal (DO) PDEs for the parabolic wave equation are derived and numerical schemes for their integration are obtained. Within the parabolic approximation, these equations are the optimal reduced-order representation of stochastic acoustic waves within the uncertain sound speed environment. The DO equations govern the propagation of the mean field, the DO modes, and their stochastic coefficients. Examples are provided for a set of idealized test cases as well as for more realistic ocean environments, and predictions are contrasted with those of other uncertainty quantification schemes. The utilization of DO equations for end-to-end uncertainty prediction within oceanographic-seabed-acoustic-sonar dynamical systems is discussed.

Energy‐optimal path planning in the coastal ocean

AbstractWe integrate data‐driven ocean modeling with the stochastic Dynamically Orthogonal (DO) level‐set optimization methodology to compute and study energy‐optimal paths, speeds, and headings for ocean vehicles in the Middle‐Atlantic Bight (MAB) region. We hindcast the energy‐optimal paths from among exact time‐optimal paths for the period 28 August 2006 to 9 September 2006. To do so, we first obtain a data‐assimilative multiscale reanalysis, combining ocean observations with implicit two‐way nested multiresolution primitive‐equation simulations of the tidal‐to‐mesoscale dynamics in the region. Second, we solve the reduced‐order stochastic DO level‐set partial differential equations (PDEs) to compute the joint probability of minimum arrival time, vehicle‐speed time series, and total energy utilized. Third, for each arrival time, we select the vehicle‐speed time series that minimize the total energy utilization from the marginal probability of vehicle‐speed and total energy. The corresponding energy‐optimal path and headings are obtained through the exact particle‐backtracking equation. Theoretically, the present methodology is PDE‐based and provides fundamental energy‐optimal predictions without heuristics. Computationally, it is 3–4 orders of magnitude faster than direct Monte Carlo methods. For the missions considered, we analyze the effects of the regional tidal currents, strong wind events, coastal jets, shelfbreak front, and other local circulations on the energy‐optimal paths. Results showcase the opportunities for vehicles that intelligently utilize the ocean environment to minimize energy usage, rigorously integrating ocean forecasting with optimal control of autonomous vehicles.

Energy-optimal path planning by stochastic dynamically orthogonal level-set optimization

A stochastic optimization methodology is formulated for computing energy-optimal paths from among time-optimal paths of autonomous vehicles navigating in a dynamic flow field. Based on partial differential equations, the methodology rigorously leverages the level-set equation that governs time-optimal reachability fronts for a given relative vehicle-speed function. To set up the energy optimization, the relative vehicle-speed and headings are considered to be stochastic and new stochastic Dynamically Orthogonal (DO) level-set equations are derived. Their solution provides the distribution of time-optimal reachability fronts and corresponding distribution of time-optimal paths. An optimization is then performed on the vehicle’s energy-time joint distribution to select the energy-optimal paths for each arrival time, among all stochastic time-optimal paths for that arrival time. Numerical schemes to solve the reduced stochastic DO level-set equations are obtained, and accuracy and efficiency considerations are discussed. These reduced equations are first shown to be efficient at solving the governing stochastic level-sets, in part by comparisons with direct Monte Carlo simulations. To validate the methodology and illustrate its accuracy, comparisons with semi-analytical energy-optimal path solutions are then completed. In particular, we consider the energy-optimal crossing of a canonical steady front and set up its semi-analytical solution using a energy-time nested nonlinear double-optimization scheme. We then showcase the inner workings and nuances of the energy-optimal path planning, considering different mission scenarios. Finally, we study and discuss results of energy-optimal missions in a wind-driven barotropic quasi-geostrophic double-gyre ocean circulation.

Error Analysis of the Dynamically Orthogonal Approximation of Time Dependent Random PDEs

In this work we discuss the dynamically orthogonal (DO) approximation of time dependent partial differential equations with random data. The approximate solution is expanded at each time instant on a time dependent orthonormal basis in the physical domain with a fixed and small number of terms. Dynamic equations are written for the evolution of the basis as well as the evolution of the stochastic coefficients of the expansion. We analyze the case of a linear parabolic equation with random data and derive a theoretical bound for the approximation error of the $S$-terms DO solution by the corresponding $S$-terms best approximation, i.e., the truncated $S$-terms Karhunen--Loeve expansion at each time instant. The bound is applicable on the largest time interval in which the best $S$-terms approximation is continuously time differentiable. Properties of the DO approximations are analyzed on simple cases of deterministic equations with random initial data. Numerical tests are presented that confirm the theoret...

On the equivalence of dynamically orthogonal and bi-orthogonal methods: Theory and numerical simulations

The Karhunen–Lòeve (KL) decomposition provides a low-dimensional representation for random fields as it is optimal in the mean square sense. Although for many stochastic systems of practical interest, described by stochastic partial differential equations (SPDEs), solutions possess this low-dimensional character, they also have a strongly time-dependent form and to this end a fixed-in-time basis may not describe the solution in an efficient way. Motivated by this limitation of standard KL expansion, Sapsis and Lermusiaux (2009) [26] developed the dynamically orthogonal (DO) field equations which allow for the simultaneous evolution of both the spatial basis where uncertainty ‘lives’ but also the stochastic characteristics of uncertainty. Recently, Cheng et al. (2013) [28] introduced an alternative approach, the bi-orthogonal (BO) method, which performs the exact same tasks, i.e. it evolves the spatial basis and the stochastic characteristics of uncertainty. In the current work we examine the relation of the two approaches and we prove theoretically and illustrate numerically their equivalence, in the sense that one method is an exact reformulation of the other. We show this by deriving a linear and invertible transformation matrix described by a matrix differential equation that connects the BO and the DO solutions. We also examine a pathology of the BO equations that occurs when two eigenvalues of the solution cross, resulting in an instantaneous, infinite-speed, internal rotation of the computed spatial basis. We demonstrate that despite the instantaneous duration of the singularity this has important implications on the numerical performance of the BO approach. On the other hand, it is observed that the BO is more stable in nonlinear problems involving a relatively large number of modes. Several examples, linear and nonlinear, are presented to illustrate the DO and BO methods as well as their equivalence.

Data Assimilation with Gaussian Mixture Models Using the Dynamically Orthogonal Field Equations. Part I: Theory and Scheme

AbstractThis work introduces and derives an efficient, data-driven assimilation scheme, focused on a time-dependent stochastic subspace that respects nonlinear dynamics and captures non-Gaussian statistics as it occurs. The motivation is to obtain a filter that is applicable to realistic geophysical applications, but that also rigorously utilizes the governing dynamical equations with information theory and learning theory for efficient Bayesian data assimilation. Building on the foundations of classical filters, the underlying theory and algorithmic implementation of the new filter are developed and derived. The stochastic Dynamically Orthogonal (DO) field equations and their adaptive stochastic subspace are employed to predict prior probabilities for the full dynamical state, effectively approximating the Fokker–Planck equation. At assimilation times, the DO realizations are fit to semiparametric Gaussian Mixture Models (GMMs) using the Expectation-Maximization algorithm and the Bayesian Information Criterion. Bayes’s law is then efficiently carried out analytically within the evolving stochastic subspace. The resulting GMM-DO filter is illustrated in a very simple example. Variations of the GMM-DO filter are also provided along with comparisons with related schemes.

Blended reduced subspace algorithms for uncertainty quantification of quadratic systems with a stable mean state

Order-reduction schemes have been used successfully for the analysis and simplification of high-dimensional systems exhibiting low-dimensional dynamics. In this work, we first focus on presenting generic limitations of order-reduction techniques in systems with stable mean state that exhibit irreducible high-dimensional features such as non-normal dynamics, wide energy spectra, or strong energy cascades between modes. The reduced-order framework that we consider to illustrate these limitations is the dynamically orthogonal (DO) field equations. This framework is applied to a series of examples with stable mean state, including a linear non-normal system, and a nonlinear triad system in various dynamical configurations. After illustrating the weaknesses and generic limitations of order reduction, we develop a novel, two-way coupled, blended approach based on the quasilinear Gaussian (QG) closure and the DO field equations. The new method (QG–DO) overcomes the limitations of its two ingredients and achieves exceptional performance in the examples described previously as well as in other configurations with strongly transient character without using any tuned or adjustable parameters.

Numerical schemes for dynamically orthogonal equations of stochastic fluid and ocean flows

The quantification of uncertainties is critical when systems are nonlinear and have uncertain terms in their governing equations or are constrained by limited knowledge of initial and boundary conditions. Such situations are common in multiscale, intermittent and non-homogeneous fluid and ocean flows. The dynamically orthogonal (DO) field equations provide an adaptive methodology to predict the probability density functions of such flows. The present work derives efficient computational schemes for the DO methodology applied to unsteady stochastic Navier–Stokes and Boussinesq equations, and illustrates and studies the numerical aspects of these schemes. Semi-implicit projection methods are developed for the mean and for the DO modes, and time-marching schemes of first to fourth order are used for the stochastic coefficients. Conservative second-order finite-volumes are employed in physical space with new advection schemes based on total variation diminishing methods. Other results include: (i) the definition of pseudo-stochastic pressures to obtain a number of pressure equations that is linear in the subspace size instead of quadratic; (ii) symmetric advection schemes for the stochastic velocities; (iii) the use of generalized inversion to deal with singular subspace covariances or deterministic modes; and (iv) schemes to maintain orthonormal modes at the numerical level. To verify our implementation and study the properties of our schemes and their variations, a set of stochastic flow benchmarks are defined including asymmetric Dirac and symmetric lock-exchange flows, lid-driven cavity flows, and flows past objects in a confined channel. Different Reynolds number and Grashof number regimes are employed to illustrate robustness. Optimal convergence under both time and space refinements is shown as well as the convergence of the probability density functions with the number of stochastic realizations.