Galerkin Projection and Numerical Integration for a Stochastic Investigation of the Viscous Burgers’ Equation: An Initial Attempt

Abstract We consider a stochastic analysis of non-linear viscous fluid flow problems with smooth and sharp gradients in stochastic space. As a representative example we consider the viscous Burgers’ equation and compare two typical intrusive and non-intrusive uncertainty quantification methods. The specific intrusive approach uses a combination of polynomial chaos and stochastic Galerkin projection. The specific non-intrusive method uses numerical integration by combining quadrature rules and the probability density functions of the prescribed uncertainties. The two methods are compared in terms of error in the estimated variance, computational efficiency and accuracy. This comparison, although not general, provide insight into uncertainty quantification of problems with a combination of sharp and smooth variations in stochastic space. It suggests that combining intrusive and non-intrusive methods could be advantageous.

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Distributed Control of the Generalized Korteweg-de Vries-Burgers Equation

Mathematical Problems in Engineering ◽

10.1155/2008/621672 ◽

2008 ◽

Vol 2008 ◽

pp. 1-19 ◽

Cited By ~ 1

Author(s):

Nejib Smaoui ◽

Rasha H. Al-Jamal

Keyword(s):

Distributed Control ◽

Feedback Linearization ◽

Burgers Equation ◽

Galerkin Projection ◽

Decomposition Procedure ◽

Control Schemes ◽

Feedback Linearization Control ◽

De Vries ◽

The Stability ◽

Model Reduction Technique

The paper deals with the distributed control of the generalized Kortweg-de Vries-Burgers equation (GKdVB) subject to periodic boundary conditions via the Karhunen-Loève (K-L) Galerkin method. The decomposition procedure of the K-L method is presented to illustrate the use of this method in analyzing the numerical simulations data which represent the solutions to the GKdVB equation. The K-L Galerkin projection is used as a model reduction technique for nonlinear systems to derive a system of ordinary differential equations (ODEs) that mimics the dynamics of the GKdVB equation. The data coefficients derived from the ODE system are then used to approximate the solutions of the GKdVB equation. Finally, three state feedback linearization control schemes with the objective of enhancing the stability of the GKdVB equation are proposed. Simulations of the controlled system are given to illustrate the developed theory.

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Evolve Filter Stabilization Reduced-Order Model for Stochastic Burgers Equation

Fluids ◽

10.3390/fluids3040084 ◽

2018 ◽

Vol 3 (4) ◽

pp. 84

Author(s):

Xuping Xie ◽

Feng Bao ◽

Clayton Webster

Keyword(s):

Stochastic Systems ◽

Burgers Equation ◽

Spatial Filter ◽

Reduced Order Model ◽

Galerkin Projection ◽

Order Model ◽

Reduced Order ◽

Numerical Result ◽

Stochastic Burgers Equation ◽

Convection Dominated

In this paper, we introduce the evolve-then-filter (EF) regularization method for reduced order modeling of convection-dominated stochastic systems. The standard Galerkin projection reduced order model (G-ROM) yield numerical oscillations in a convection-dominated regime. The evolve-then-filter reduced order model (EF-ROM) aims at the numerical stabilization of the standard G-ROM, which uses explicit ROM spatial filter to regularize various terms in the reduced order model (ROM). Our numerical results are based on a stochastic Burgers equation with linear multiplicative noise. The numerical result shows that the EF-ROM is significantly better than G-ROM.

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On the numerical integration of Burgers' equation by stochastic simulation methods

Computer Physics Communications ◽

10.1016/0010-4655(93)90004-v ◽

1993 ◽

Vol 77 (2) ◽

pp. 207-218 ◽

Cited By ~ 4

Author(s):

Heinz-Peter Breuer ◽

Francesco Petruccione

Keyword(s):

Numerical Integration ◽

Stochastic Simulation ◽

Burgers Equation ◽

Simulation Methods

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The orbit of Pluto over a long interval of time

Symposium - International Astronomical Union ◽

10.1017/s0074180900105509 ◽

1966 ◽

Vol 25 ◽

pp. 227-229 ◽

Cited By ~ 1

Author(s):

D. Brouwer

Keyword(s):

Periodic Orbit ◽

Numerical Integration ◽

Restricted Problem ◽

Three Dimensional ◽

The Third ◽

Circular Restricted Problem ◽

Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

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