A Higher Order Ensemble Simulation Algorithm for Fluid Flows

Jiang, Nan

doi:10.1007/s10915-014-9932-z

Cited by 49 publications

(44 citation statements)

References 35 publications

(42 reference statements)

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“…For instance, the method is not extensible to the most commonly used Crank-Nicolson scheme. Making use of a special combination of a second-order in time backward difference formula and an explicit second-order Adams-Bashforth treatment of the nonlinear term, a second-order accurate in time ensemble method was developed in [19].Another second-order ensemble method with improved accuracy is presented in [20]. The ensemble algorithm was further used in [21] to model turbulence.…”

Section: Previous Work On Ensemble Algorithmsmentioning

confidence: 99%

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An Ensemble-Proper Orthogonal Decomposition Method for the Nonstationary Navier--Stokes Equations

Gunzburger¹,

Jiang²,

Schneier³

2017

SIAM J. Numer. Anal.

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Abstract. The definition of partial differential equation (PDE) models usually involves a set of parameters whose values may vary over a wide range. The solution of even a single set of parameter values may be quite expensive. In many cases, e.g., optimization, control, uncertainty quantification, and other settings, solutions are needed for many sets of parameter values. We consider the case of the time-dependent Navier-Stokes equations for which a recently developed ensemble-based method allows for the efficient determination of the multiple solutions corresponding to many parameter sets. The method uses the average of the multiple solutions at any time step to define a linear set of equations that determines the solutions at the next time step. To significantly further reduce the costs of determining multiple solutions of the Navier-Stokes equations, we incorporate a proper orthogonal decomposition (POD) reduced-order model into the ensemble-based method. The stability and convergence results for the ensemble-based method are extended to the ensemble-POD approach. Numerical experiments are provided that illustrate the accuracy and efficiency of computations determined using the new approach.Key words. Ensemble methods, proper orthogonal decomposition, reduced-order models, Navier-Stokes equations.1. Introduction. Computing an ensemble of solutions of fluid flow equations for a set of parameters or initial/boundary conditions for, e.g., quantifying uncertainty or sensitivity analyses or to make predictions, is a common procedure in many engineering and geophysical applications. One common problem faced in these calculations is the excessive cost in terms of both storage and computing time. Thanks to recent rapid advances in parallel computing as well as intensive research in ensemble-based data assimilation, it is now possible, in certain settings, to obtain reliable ensemble predictions using only a small set of realizations. Successful methods that are currently used to generate perturbations in initial conditions include the Bred-vector method, [30], the singular vector method, [5], and the ensemble transform Kalman filter, [4]. Despite all these efforts, the current level of available computing power is still insufficient to perform high-accuracy ensemble computations for applications that deal with large spatial scales such as numerical weather prediction. In such applications, spatial resolution is often sacrificed to reduce the total computational time. For these reasons the development of efficient methods that allow for fast calculation of flow ensembles at a sufficiently fine spatial resolution is of great practical interest and significance.Only recently, a first step was taken in [22,23] where a new algorithm was proposed for computing an ensemble of solutions of the time-dependent Navier-Stokes equations (NSE) with different initial condition and/or body forces. At each time step, the new method employs the same coefficient matrix for all ensemble members. This reduces the problem of solving multi...

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Section: Previous Work On Ensemble Algorithmsmentioning

confidence: 99%

“…For single Navier-Stokes solves, there is existing in vast literature in this regard, but, in the ensemble setting, regularization has barely been studied. The only existing works are in [19,23]. The study of regularization methods in the ensemble and ensemble-POD setting is a focus of our current research.…”

Section: Examplementioning

confidence: 99%

An Ensemble-Proper Orthogonal Decomposition Method for the Nonstationary Navier--Stokes Equations

Gunzburger¹,

Jiang²,

Schneier³

2017

SIAM J. Numer. Anal.

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“…Stability tests. Next, we check the stability of our algorithm by considering the problem of a flow between two offset circles [11,12,14,15]. The domain is a disk with a smaller off-center obstacle inside.…”

Section: Convergence Testmentioning

confidence: 99%

A Second-Order Time-Stepping Scheme for Simulating Ensembles of Parameterized Flow Problems

Gunzburger

Jiang

Wang

2017

Computational Methods in Applied Mathematics

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Abstract. We consider settings for which one needs to perform multiple flow simulations based on the Navier-Stokes equations, each having different values for the physical parameters and/or different initial condition data, boundary conditions data, and/or forcing functions. For such settings, we propose a second-order time accurate ensemble-based method that to simulate the whole set of solutions, requires, at each time step, the solution of only a single linear system with multiple righthand-side vectors. Rigorous analyses are given proving the conditional stability and error estimates for the proposed algorithm. Numerical experiments are provided that illustrate the analyses.Key words. Navier-Stokes equations, parameterized flow, ensemble method MSC (2010). 65M60, 76D051. Introduction. Many computational fluid dynamics applications require multiple simulations of a flow under different input conditions. For example, the ensemble Kalman filter approach used in data assimilation first simulates a forward model a large number of times by perturbing either the initial condition data, boundary condition data, or uncertain parameters, then corrects the model based on the model forecasts and observational data. A second example is the construction of low-dimensional surrogates for solutions of partial differential equation (PDE) such as sparse-grid interpolants or proper orthogonal decomposition approximations for which one has to first obtain expensive approximations of solutions corresponding to several parameter samples. Another example is sensitivity analyses of solutions for which one often has to determine approximate solutions for a number of perturbed inputs such as the values of certain physical parameters. In this paper, we consider such applications and develop a second-order time-stepping scheme for efficiently simulating an ensemble of flows. In particular, we consider the setting in which one wishes to determine the PDE solutions for several different values of the physical parameters and with several different choices of initial condition and boundary condition data and forcing functions appearing in the PDE model.The ensemble algorithm we use was first developed in [11] to find a set of J solutions of the Navier-Stokes equations (NSE) subject to different initial condition and forcing functions. The main idea is that, based on the introduction of an ensemble average and a special semi-implicit time discretization, the discrete systems for the multiple flow simulations share a common coefficient matrix. Thus, instead of solving J linear system with J right-hand sides (RHS), one only need solve one linear system with J RHS. This leads to great computational saving in linear solvers when either the LU factorization (for small-scale systems) or a block iterative algorithm (for large-

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“…First, we verify predicted convergence rates on a 2d test problem with known analytical solution. We also compare accuracy of (En‐BlendedBDF) with that of the previously studied (En‐BDF2AB2) method (see ). The (En‐BDF2AB2) method is given by

\begin{matrix} left \\ \frac{3 u_{j}^{n + 1} - 4 u_{j}^{n} + u_{j}^{n - 1}}{2 Δ t} + < u >^{n} \cdot \nabla u_{j}^{n + 1} (En‐BlendedBDF) \\ + u_{j}^{' n} \cdot \nabla true(2 u_{j}^{n} - u_{j}^{n - 1} true) + \nabla p_{j}^{n + 1} - ν Δ u_{j}^{n + 1} = f_{j}^{n + 1}, \\ \nabla \cdot u_{j}^{n + 1} = 0. \end{matrix}

Next, we test the ability of the method to simulate high Reynolds number, complex flows.…”

Section: Numerical Experimentsmentioning

confidence: 99%

A second‐order ensemble method based on a blended backward differentiation formula timestepping scheme for time‐dependent Navier–Stokes equations

Jiang

2016

Numerical Methods Partial

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We present a second‐order ensemble method based on a blended three‐step backward differentiation formula (BDF) timestepping scheme to compute an ensemble of Navier–Stokes equations. Compared with the only existing second‐order ensemble method that combines the two‐step BDF timestepping scheme and a special explicit second‐order Adams–Bashforth treatment of the advection term, this method is more accurate with nominal increase in computational cost. We give comprehensive stability and error analysis for the method. Numerical examples are also provided to verify theoretical results and demonstrate the improved accuracy of the method. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 34–61, 2017

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A Higher Order Ensemble Simulation Algorithm for Fluid Flows

Cited by 49 publications

References 35 publications

An Ensemble-Proper Orthogonal Decomposition Method for the Nonstationary Navier--Stokes Equations

An Ensemble-Proper Orthogonal Decomposition Method for the Nonstationary Navier--Stokes Equations

A Second-Order Time-Stepping Scheme for Simulating Ensembles of Parameterized Flow Problems

A second‐order ensemble method based on a blended backward differentiation formula timestepping scheme for time‐dependent Navier–Stokes equations

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