| 1. |
- Nordström, Jan, et al.
(författare)
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Boundary Conditions for Hyperbolic Systems of Equations on Curved Domains
- 2014
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Our focus in this paper is on the fundamental system of partial differential equation with boundary conditions (the continuous problem) that all types of numerical methods must respect. First, a constant coefficient hyperbolic system of equations which turns into a variable coefficient system of equations by transforming to a non-cartesian domain is considered. We discuss possible formulations of time-dependent boundary conditions leading to well-posed or strongly well-posed problems. Next, we re-use the previous theoretical derivations for the problem with boundary conditions applied at the wrong position and/or with an incorrect normal (a typical result with a less than perfect mesh generator). Possible error sources are discussed and a crude error estimate is derived.
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| 2. |
- Nordström, Jan, 1953-, et al.
(författare)
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Robust Design of Initial Boundary Value Problems
- 2019
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Ingår i: Uncertainty Management for Robust Industrial Design in Aeronautics. - Cham : Springer. - 9783319777672 - 9783319777665 ; , s. 463-478
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Bokkapitel (refereegranskat)abstract
- We study hyperbolic and incompletely parabolic systems with stochastic boundary and initial data. Estimates of the variance of the solution are presented both analytically and numerically. It is shown that one can reduce the variance for a given input, with a specific choice of boundary condition. The technique is applied to the Maxwell, Euler, and Navier–Stokes equations. Numerical calculations corroborate the theoretical conclusions.
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| 3. |
- Nordström, Jan, et al.
(författare)
-
Variance reduction through robust design of boundary conditions for stochastic hyperbolic systems of equations
- 2014
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We consider a hyperbolic system in one space dimension with uncertainty in the boundary and initial data. Our aim is to show that di erent boundary conditions gives different convergence rates of the variance of the solution. This means that we can with the same knowledge of data get a more or less accurate description of the uncertainty in the solution. A variety of boundary conditions are compared and both analytical and numerical estimates of the variance of the solution is presented. As an application, we study the effect of this technique on a subsonic outow boundary for the Euler equations.
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| 4. |
- Nordström, Jan, et al.
(författare)
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Variance reduction through robust design of boundary conditions for stochastic hyperbolic systems of equations
- 2015
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 282, s. 1-22
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Tidskriftsartikel (refereegranskat)abstract
- We consider a hyperbolic system with uncertainty in the boundary and initial data. Our aim is to show that different boundary conditions gives different convergence rates of the variance of the solution. This means that we can with the same knowledge of data get a more or less accurate description of the uncertainty in the solution. A variety of boundary conditions are compared and both analytical and numerical estimates of the variance of the solution is presented. As applications, we study the effect of this technique on Maxwell's equations as well as on a subsonic outflow boundary for the Euler equations.
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| 5. |
- Wahlsten, Markus, 1986-, et al.
(författare)
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An efficient hybrid method for uncertainty quantification
- 2018
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- A technique for coupling an intrusive and non-intrusive uncertainty quantification method is proposed. The intrusive approach uses a combination of polynomial chaos and stochastic Galerkin projection. The non-intrusive method uses numerical integration by combining quadrature rules and the probability density functions of the prescribed uncertainties. A strongly stable coupling procedure between the two methods at an interface is constructed. The efficiency of the hybrid method is exemplified using a hyperbolic system of equations, and verified by numerical experiments.
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| 6. |
- Wahlsten, Markus, 1986-, et al.
(författare)
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An efficient hybrid method for uncertainty quantification
- 2022
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Ingår i: BIT Numerical Mathematics. - : SPRINGER. - 0006-3835 .- 1572-9125. ; 62, s. 607-629
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Tidskriftsartikel (refereegranskat)abstract
- A technique for coupling an intrusive and non-intrusive uncertainty quantification method is proposed. The intrusive approach uses a combination of polynomial chaos and stochastic Galerkin projection. The non-intrusive method uses numerical integration by combining quadrature rules and the probability density functions of the prescribed uncertainties. A stable coupling procedure between the two methods at an interface is constructed. The efficiency of the hybrid method is exemplified using hyperbolic systems of equations, and verified by numerical experiments.
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| 7. |
- Wahlsten, Markus, et al.
(författare)
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An Investigation of Uncertainty due to Stochastically Varying Geometry : An Initial Study
- 2015
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Konferensbidrag (övrigt vetenskapligt/konstnärligt)abstract
- We study hyperbolic problems with uncertain stochastically varying geometries. Our aim is to investigate how the stochastically varying uncertainty in the geometry affects the solution of the partial differential equation in terms of the mean and variance of the solution. The problem considered is the two dimensional advection equation on a general domain, which is transformed using curvilinear coordinates to a unit square. The numerical solution is computed using a high order finite difference formulation on summation-by-parts form with weakly imposed boundary conditions. The statistics of the solution are computed nonintrusively using quadrature rules given by the probability density function of the random variable.We prove that the continuous problem is strongly well-posed and that the semi-discrete problem is strongly stable. Numerical calculations using the method of manufactured solution verify the accuracy of the scheme and the statistical properties of the solution are discussed.
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| 8. |
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| 9. |
- Wahlsten, Markus, 1986-, et al.
(författare)
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Galerkin projection and numerical integration for a stochastic investigation of the viscous Burgers equation : An initial attempt
- 2019
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Ingår i: Numerical Mathematics and Advanced Applications ENUMATH 2017. ENUMATH 2017. - Cham : Springer. - 9783319964140 - 9783319964157 ; , s. 1005-1013
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Bokkapitel (refereegranskat)abstract
- We consider a stochastic analysis of the non-linear viscous Burgers’ equation and focus on the comparison between intrusive and non-intrusive uncertainty quantification methods. The intrusive approach uses a combination of polynomial chaos and stochastic Galerkin projection. The non-intrusive method uses numerical integration by combining quadrature rules and the probability density functions of the prescribed uncertainties. The two methods are applied to a provably stable formulation of the viscous Burgers’ equation, and compared. As measures of comparison: variance size, computational efficiency and accuracy are used.
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| 10. |
- Wahlsten, Markus, 1986-, et al.
(författare)
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On Stochastic Investigation of Flow Problems Using the Viscous Burgers’ Equation as an Example
- 2019
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Ingår i: Journal of Scientific Computing. - : Springer Science and Business Media LLC. - 0885-7474 .- 1573-7691. ; 81:2, s. 1111-1117
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Tidskriftsartikel (refereegranskat)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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