Boundary procedures for the time-dependent Burgers' equation under uncertainty

Pettersson, Per; Nordström, Jan; Iaccarino, Gianluca

doi:10.1016/s0252-9602(10)60061-6

Cited by 10 publications

(5 citation statements)

References 13 publications

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“…Other areas of application include: conjugate-heat transfer [56,57]; uncertainty quantification [58,59,60]; magnetic induction [61,62,63]; Wave propagation [64,65,66,67,68]. Other interesting examples include, DNS of the interaction between a turbulent boundary layer and a fluttering panel [69]; and, computation of acoustically-excited flow through a circular orifice backed by a hexagonal cavity [70].…”

Section: Applicationmentioning

confidence: 99%

Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations

2014

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Summation-by-parts (SBP) operators have a number of properties that make them an attractive option for higher-order spatial discretizations of partial differential equations. In particular, they enable the derivation of higher-order boundary closures leading to provable time stability. When implemented on multi-block structured meshes in conjunction with simultaneous approximation terms (SATs)-penalty terms that impose boundary and interblock-coupling conditions in a weak sense-they offer additional properties of value, even for second-order accurate schemes and steady problems. For example, they involve low communication overhead for efficient parallel algorithms and relax the continuity requirements of both the mesh and the solution across block interfaces. This paper provides a brief history of seminal contributions to, and applications of, SBP-SAT methods followed by a description of their properties and a methodology for deriving SBP operators for first derivatives and second derivatives with variable coefficients. A procedure for deriving SATs is also provided. Practical aspects are discussed, including artificial dissipation, transformation to curvilinear coordinates, and application to the Navier-Stokes equations. Recent developments are reviewed, including a variational interpretation, the connection to quadrature rules, functional superconvergence, error estimates, and dual consistency. Finally, the connection to quadrature rules is exploited to provide a generalization of the SBP concept to a broader class of operators, enabling a unification and rigorous development of SATs for operators such as nodal-based pseudo-spectral and some discontinuous Galerkin operators.

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Section: Applicationmentioning

confidence: 99%

Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations

2014

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“…As noted in the classic 1982 paper by Fletcher [26], Burgers' equation is often used as a test case for numerical methods to illustrate accuracy and convergence of a particular scheme. This is still true today as illustrated by the recent references (see [23,32,35,38,44,48] and the references therein). It is interesting to note that although Burgers' equation has served as a great test case for numerous numerical algorithms, almost all papers along this line use Burgers' equation with Dirichlet boundary conditions.…”

Section: Introductionmentioning

confidence: 88%

Numerical Approximations of the Dynamical System Generated by Burgers’ Equation with Neumann–Dirichlet Boundary Conditions

Allen¹,

Burns²,

Gilliam³

2013

ESAIM: M2AN

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Using Burgers' equation with mixed Neumann-Dirichlet boundary conditions, we highlight a problem that can arise in the numerical approximation of nonlinear dynamical systems on computers with a finite precision floating point number system. We describe the dynamical system generated by Burgers' equation with mixed boundary conditions, summarize some of its properties and analyze the equilibrium states for finite dimensional dynamical systems that are generated by numerical approximations of this system. It is important to note that there are two fundamental differences between Burgers' equation with mixed Neumann-Dirichlet boundary conditions and Burgers' equation with both Dirichlet boundary conditions. First, Burgers' equation with homogenous mixed boundary conditions on a finite interval cannot be linearized by the Cole-Hopf transformation. Thus, on finite intervals Burgers' equation with a homogenous Neumann boundary condition is truly nonlinear. Second, the nonlinear term in Burgers' equation with a homogenous Neumann boundary condition is not conservative. This structure plays a key role in understanding the complex dynamics generated by Burgers' equation with a Neumann boundary condition and how this structure impacts numerical approximations. The key point is that, regardless of the particular numerical scheme, finite precision arithmetic will always lead to numerically generated equilibrium states that do not correspond to equilibrium states of the Burgers' equation. In this paper we establish the existence and stability properties of these numerical stationary solutions and employ a bifurcation analysis to provide a detailed mathematical explanation of why numerical schemes fail to capture the correct asymptotic dynamics. We extend the results in [E. Allen,

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“…For example, the principal stress axes were computed and perturbed to estimate the effect of structural uncertainty of turbulence models on flow realizations. The effects of uncertain boundary conditions have been examined using the Burgers' equation and the Navier–Stokes equations . Bayesian estimation of the effects of the combustion models in LES has also been carried out , while in cold flow isotropic turbulence, Lucor et al assumed a distribution for the constant in the Smagorinsky model following a polynomial chaos expansion (PCE), which was sampled to compute the PCE of different scales of turbulent motion.…”

Section: Introductionmentioning

confidence: 99%

Uncertainty quantification in LES of channel flow

Safta

Blaylock

Templeton

et al. 2016

Numerical Methods in Fluids

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Summary In this paper, we present a Bayesian framework for estimating joint densities for large eddy simulation (LES) sub‐grid scale model parameters based on canonical forced isotropic turbulence direct numerical simulation (DNS) data. The framework accounts for noise in the independent variables, and we present alternative formulations for accounting for discrepancies between model and data. To generate probability densities for flow characteristics, posterior densities for sub‐grid scale model parameters are propagated forward through LES of channel flow and compared with DNS data. Synthesis of the calibration and prediction results demonstrates that model parameters have an explicit filter width dependence and are highly correlated. Discrepancies between DNS and calibrated LES results point to additional model form inadequacies that need to be accounted for. Copyright © 2016 John Wiley & Sons, Ltd.

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Boundary procedures for the time-dependent Burgers' equation under uncertainty

Cited by 10 publications

References 13 publications

Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations

Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations

Numerical Approximations of the Dynamical System Generated by Burgers’ Equation with Neumann–Dirichlet Boundary Conditions

Uncertainty quantification in LES of channel flow

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