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A Third Order Hierarchical Basis WENO Interpolation for Sparse Grids with Application to Conservation Laws with Uncertain Data

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Oliver Kolb at University of Mannheim
Oliver Kolb
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Abstract and Figures

In this paper, we introduce a third order hierarchical basis WENO interpolation, which possesses similar accuracy and stability properties as usual WENO interpolations. The main motivation for the hierarchical approach is the direct applicability on sparse grids. This is for instance of large practical interest in the numerical solution of conservation laws with uncertain data, where discontinuities in the physical domain often carry over to the (potentially high-dimensional) stochastic domain. For this, we apply the introduced hierarchical basis WENO interpolation within a non-intrusive collocation method and present first results on 2- and 3-dimensional sparse grids.
Illustration of the index notation
Illustration of the index notation
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Results for Burgers’ equation with three uncertain parameters; x∈[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in [0,1]$$\end{document}, t=0.15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=0.15$$\end{document}, ξ∈[0.2,0.5]×[-0.25,0.2]×[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi \in [0.2,0.5]\times [-0.25,0.2]\times [0,1]$$\end{document}
Results for Burgers’ equation with three uncertain parameters; x∈[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in [0,1]$$\end{document}, t=0.15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=0.15$$\end{document}, ξ∈[0.2,0.5]×[-0.25,0.2]×[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi \in [0.2,0.5]\times [-0.25,0.2]\times [0,1]$$\end{document}
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J Sci Comput (2018) 74:1480–1503
https://doi.org/10.1007/s10915-017-0503-y
A Third Order Hierarchical Basis WENO Interpolation
for Sparse Grids with Application to Conservation Laws
with Uncertain Data
Oliver Kolb1
Received: 25 May 2016 / Revised: 8 June 2017 / Accepted: 12 July 2017 / Published online: 19 July 2017
© Springer Science+Business Media, LLC 2017
Abstract In this paper, we introduce a third order hierarchical basis WENO interpolation,
which possesses similar accuracy and stability properties as usual WENO interpolations. The
main motivation for the hierarchical approach is the direct applicability on sparse grids. This
is for instance of large practical interest in the numerical solution of conservation laws with
uncertain data, where discontinuities in the physical domain often carry over to the (potentially
high-dimensional) stochastic domain. For this, we apply the introduced hierarchical basis
WENO interpolation within a non-intrusive collocation method and present first results on
2- and 3-dimensional sparse grids.
Keywords Weighted essentially nonoscillatory interpolation ·Sparse grid ·Uncertainty
quantification
1 Introduction
In recent years, there has been a rapid development of methods for uncertainty quantification
in the solution of partial differential equations (PDEs) with uncertain data. Here, one is typi-
cally interested in the so-called response surface as well as statistics of the solution. Common
approaches are stochastic Galerkin methods/polynomial chaos [9,10,25–27], stochastic col-
location [5,6], Monte Carlo [17,18,22] and sparse grid quadrature [19]. Except for Monte
Carlo methods, which can only be applied for the computation of statistics, all other methods
at least implicitly require certain smoothness assumptions with respect to the considered
random variables. Now, it is well known that in the context of hyperbolic PDEs the solution
of single deterministic problems may contain discontinuities after finite time even for arbi-
trary smooth data, and discontinuities in the physical domain often directly carry over to the
BOliver Kolb
kolb@uni-mannheim.de
1University of Mannheim, Mannheim, Germany
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... Multiresolution analysis has been successfully used to develop sparse-grid discontinuous Galerkin methods in [10,11,26]. To develop sparse-grid MRWENO methods along this way, we will use the WENO reconstruction procedure based on the hierarchical basis in [14] and incorporate high-order MRWENO approximations in it. Due to the hierarchical structure of the reconstruction procedure in [14], it can be naturally applied to sparse grids, and a tensor product approach can be used for multidimensional problems. ...
... To develop sparse-grid MRWENO methods along this way, we will use the WENO reconstruction procedure based on the hierarchical basis in [14] and incorporate high-order MRWENO approximations in it. Due to the hierarchical structure of the reconstruction procedure in [14], it can be naturally applied to sparse grids, and a tensor product approach can be used for multidimensional problems. Furthermore, this kind of hierarchical basis WENO methods on sparse grids will be promising to be combined with adaptive approaches, since the hierarchical basis or the hierarchical structure in the schemes provide a convenient way to design error indicators in adaptive methods, as in [1,11]. ...
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Essentially Non-oscillatory Stencil Selection and Subcell Resolution in Uncertainty Quantification
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The essentially non-oscillatory stencil selection and subcell resolution robustness concepts from finite volume methods for computational fluid dynamics are extended to uncertainty quantification for the reliable approximation of discontinuities in stochastic computational problems. These two robustness principles are introduced into the simplex stochastic collocation uncertainty quantification method, which discretizes the probability space using a simplex tessellation of sampling points and piecewise higher-degree polynomial interpolation. The essentially non-oscillatory stencil selection obtains a sharper refinement of discontinuities by choosing the interpolation stencil with the highest polynomial degree from a set of candidate stencils for constructing the local response surface approximation. The subcell resolution approach achieves a genuinely discontinuous representation of random spatial discontinuities in the interior of the simplexes by resolving the discontinuity location in the probability space explicitly and by extending the stochastic response surface approximations up to the predicted discontinuity location. The advantages of the presented approaches are illustrated by the results for a step function, the linear advection equation, a shock tube Riemann problem, and the transonic flow over the RAE 2822 airfoil.
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This paper deals with the formulation of a semi-intrusive (SI) method allowing the computation of statistics of linear and non linear PDEs solutions. This method shows to be very efficient to deal with probability density function of whatsoever form, long-term integration and discontinuities in stochastic space.Given a stochastic PDE where randomness is defined on Ω, starting from (i) a description of the solution in term of a space variables, (ii) a numerical scheme defined for any event ω∈Ωω∈Ω and (iii) a (family) of random variables that may be correlated, the solution is numerically described by its conditional expectancies of point values or cell averages and its evaluation constructed from the deterministic scheme. One of the tools is a tessellation of the random space as in finite volume methods for the space variables. Then, using these conditional expectancies and the geometrical description of the tessellation, a piecewise polynomial approximation in the random variables is computed using a reconstruction method that is standard for high order finite volume space, except that the measure is no longer the standard Lebesgue measure but the probability measure. This reconstruction is then used to formulate a scheme on the numerical approximation of the solution from the deterministic scheme. This new approach is said semi-intrusive because it requires only a limited amount of modification in a deterministic solver to quantify uncertainty on the state when the solver includes uncertain variables.The effectiveness of this method is illustrated for a modified version of Kraichnan–Orszag three-mode problem where a discontinuous pdf is associated to the stochastic variable, and for a nozzle flow with shocks. The results have been analyzed in terms of accuracy and probability measure flexibility. Finally, the importance of the probabilistic reconstruction in the stochastic space is shown up on an example where the exact solution is computable, the viscous Burgers equation.
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