uncertainty quantification and numerical methods for conservation
TRANSCRIPT
ACTAUNIVERSITATIS
UPSALIENSISUPPSALA
2013
Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1008
Uncertainty Quantificationand Numerical Methods forConservation Laws
PER PETTERSSON
ISSN 1651-6214ISBN 978-91-554-8569-6urn:nbn:se:uu:diva-188348
Dissertation presented at Uppsala University to be publicly examined in Room 2446,Polacksbacken, Lägerhyddsvägen 2D, Uppsala, Friday, February 8, 2013 at 10:15 for thedegree of Doctor of Philosophy. The examination will be conducted in English.
AbstractPettersson, P. 2013. Uncertainty Quantification and Numerical Methods for ConservationLaws. Acta Universitatis Upsaliensis. Digital Comprehensive Summaries of UppsalaDissertations from the Faculty of Science and Technology 1008. 39 pp. Uppsala.ISBN 978-91-554-8569-6.
Conservation laws with uncertain initial and boundary conditions are approximated usinga generalized polynomial chaos expansion approach where the solution is represented as ageneralized Fourier series of stochastic basis functions, e.g. orthogonal polynomials or wavelets.The stochastic Galerkin method is used to project the governing partial differential equationonto the stochastic basis functions to obtain an extended deterministic system.
The stochastic Galerkin and collocation methods are used to solve an advection-diffusionequation with uncertain viscosity. We investigate well-posedness, monotonicity and stability forthe stochastic Galerkin system. High-order summation-by-parts operators and weak impositionof boundary conditions are used to prove stability. We investigate the impact of the total spatialoperator on the convergence to steady-state.
Next we apply the stochastic Galerkin method to Burgers' equation with uncertain boundaryconditions. An analysis of the truncated polynomial chaos system presents a qualitativedescription of the development of the solution over time. An analytical solution is derivedand the true polynomial chaos coefficients are shown to be smooth, while the correspondingcoefficients of the truncated stochastic Galerkin formulation are shown to be discontinuous. Wediscuss the problematic implications of the lack of known boundary data and possible ways ofimposing stable and accurate boundary conditions.
We present a new fully intrusive method for the Euler equations subject to uncertainty basedon a Roe variable transformation. The Roe formulation saves computational cost compared tothe formulation based on expansion of conservative variables. Moreover, it is more robust andcan handle cases of supersonic flow, for which the conservative variable formulation fails toproduce a bounded solution. A multiwavelet basis that can handle discontinuities in a robustway is used.
Finally, we investigate a two-phase flow problem. Based on regularity analysis of thegeneralized polynomial chaos coefficients, we present a hybrid method where solution regionsof varying smoothness are coupled weakly through interfaces. In this way, we couple smoothsolutions solved with high-order finite difference methods with non-smooth solutions solvedfor with shock-capturing methods.
Keywords: uncertainty quantification, polynomial chaos, stochastic Galerkin methods,conservation laws, hyperbolic problems, finite difference methods, finite volume methods
Per Pettersson, Uppsala University, Department of Information Technology, Division ofScientific Computing and Numerical Analysis, Box 337, SE-751 05 Uppsala, Sweden.
© Per Pettersson 2013
ISSN 1651-6214ISBN 978-91-554-8569-6urn:nbn:se:uu:diva-188348 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-188348)
List of papers
This thesis is based on the following papers, which are referred to in the textby their Roman numerals.
I P. Pettersson, G. Iaccarino and J. Nordström. Numerical analysis of theBurgers’ equation in the presence of uncertainty, J. Comput. Phys.228:8394-8412, 2009.Contributions: The ideas were developed in close collaboration between theauthors. The author of this thesis performed part of the analysis, implementedthe numerical methods and performed all the computations. The manuscriptwas written in close cooperation between the authors.
II P. Pettersson, G. Iaccarino and J. Nordström. Boundary procedures forthe time-dependent Burgers’ equation under uncertainty, ActaMathematica Scientia, 30(2):539-550, 2010.Contributions: The author of this thesis performed part of the analysis and allthe computations. The manuscript was written in close cooperation betweenthe authors.
III P. Pettersson, A. Doostan and J. Nordström. On Stability andMonotonicity Requirements of Discretized Stochastic ConservationLaws with Random Viscosity, Technical report 2012-028, Departmentof Information Technology, Uppsala University, 2012. Submitted toComputer Methods in Applied Mechanics and Engineering.Contributions: The ideas were developed in close collaboration between theauthors. The author of this thesis performed part of the analysis and all thecomputations. The manuscript was written in close cooperation between theauthors.
IV P. Pettersson, G. Iaccarino and J. Nordström. A stochastic Galerkinmethod for the Euler equations with Roe variable transformation,Technical report 2012-033, Department of Information Technology,Uppsala University, 2012. Submitted.Contributions: The author of this thesis performed most of the analysis,designed the numerical method and performed the computations. Themanuscript was written in close cooperation between the authors.
V P. Pettersson, G. Iaccarino and J. Nordström. An intrusive hybridmethod for discontinuous two-phase flow under uncertainty, Technical
report 2012-035, Department of Information Technology, UppsalaUniversity, 2012.Contributions: The ideas were developed in close collaboration between theauthors. The author of this thesis implemented the methods, performed mostof the analysis and all the computations and had the main responsibility forpreparing the manuscript.
Reprints were made with permission from the publishers.
Related work
Although neither explicitly discussed in the comprehensive summary, nor in-cluded among the papers of the thesis, the following conference proceeding isclosely related to Paper I and II.
• P. Pettersson, Q. Abbas, G. Iaccarino, and J. Nordström. Efficiency ofshock capturing schemes for Burgers’ equation with boundary uncer-tainty. In G. Kreiss, P. Lötstedt, A. Målqvist, M. Neytcheva, editors,Numerical Mathematics and Advanced Applications: 2009, 737-745,Springer, Berlin, 2010.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1 Uncertainty in physical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Random field representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Karhunen-Loève expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Generalized chaos expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Generalized polynomial chaos expansion . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Haar wavelet expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.3 Multiwavelet expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Polynomial chaos methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1 Intrusive methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.1 Stochastic Galerkin methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1.2 Semi-intrusive methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Non-intrusive methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.1 Stochastic collocation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2.2 Spectral projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.3 Stochastic multi-elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.1 Summation-by-parts operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1.1 Artificial dissipation operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.1.2 Multiple spatial domains with weak SBP-SAT
coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 Shock capturing methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2.1 MUSCL scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2.2 HLL Riemann solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5 Summary of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.5 Paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6 Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
7 Stokastiska Galerkinmetoder för konserveringslagar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
8 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1. Introduction
1.1 Uncertainty in physical systemsIn many physical problems, data is limited in quality and quantity by variabil-ity, bias in the measurements and by limitations to the extent measurementsare possible to perform. When we attempt to solve the problem at hand numer-ically, we must account for those limitations, and in addition we must identifythe possible limitations of the numerical techniques and phenomenologicalmodels employed.
In a general sense, we distinguish between errors and uncertainty simply bysaying that errors are recognizable deficiencies not due to lack of knowledge,whereas uncertainties are potential and directly related to lack of knowledge[44]. This definition clearly identifies errors as deterministic quantities anduncertainties as stochastic in nature; uncertainty estimation and quantificationis, therefore, typically treated within a probabilistic framework.
Uncertainty quantification is also a fundamental step towards validation andcertification of numerical methods to be used for critical decisions. Fieldsof application of uncertainty quantification include but are not limited to tur-bulence, climatology [48], turbulent combustion [49], flow in porous media[25, 16], fluid mixing [74] and electromagnetics [15].
An example of the need for uncertainty quantification in applications relatedto methods and problems studied here is the investigation of the aerodynamicstability properties of an airfoil. Uncertainty in physical parameters such asstructural frequency and initial pitch angle, affect the probability of limit cycleoscillations. One approach in particular, the polynomial chaos method, hasbeen used to obtain a statistical characterization of the stability limits and tocalculate the risk for system failure [68, 8]; this approach will be studied indetail throughout this thesis.
The sources of uncertainty that we consider in this thesis are impreciseknowledge of the input data, e.g. uncertainty due to finite sample sizes ormeasurement errors. This results in numerical models that are subject to un-certainty in boundary or initial conditions, model parameter values and evenin the geometry of the physical domain of the problem (input uncertainty).Uncertainty quantification in the sense it will be used here is concerned withthe propagation of input uncertainty through the numerical model in order toclearly identify and quantify the uncertainty in the output quantities of interest.
Without going into the details of how to transform a set of data into proba-bility distributions of the input variables [26], the starting point will be a partial
9
differential equation formulation where parameters and initial and boundaryconditions are uncertain but determined in terms of probability distributions.Random variables are used to parametrize the uncertainty in the input data. Aspectral series representation, the generalized chaos series expansion, is thenused to represent the solution to the problem of interest.
The one-dimensional test problems that will be investigated here are ev-idently subject to modeling error, should we like to use them as models ofreal-world phenomena. For instance, we disregard viscous forces in the flowproblems and ignore reflections from the wall of the shock tube, treating itas a purely one-dimensional problem. Thus, we do not account for epistemicuncertainty, i.e. uncertainty in the physical and mathematical models them-selves. In real-world problems, this would be an important point. If the con-ceptual model is erroneous, for instance due to an incompressibility assump-tion for a case of high Mach number flow, then there is clearly no sense in asolution no matter the degree of accuracy of the representation of variabilityin the input parameters [50].
There are several approaches to propagate the input uncertainty in numeri-cal simulations; the simplest one is the Monte Carlo method where a vast num-ber of simulations are performed to compute the output statistics. Converselyin the polynomial chaos approach, the solution is expressed as a truncated se-ries and only one simulation is performed. The dimension of the resultingsystem of equations grows with the number of the terms retained in the se-ries (the order of the polynomial chaos expansion) and the dimension of thestochastic input.
An increased number of Monte Carlo simulations implies a solution withbetter converged statistics; on the other hand, in the polynomial chaos ap-proach, one single simulation is sufficient to obtain a complete statistical char-acterization of the solution. However, the accuracy of this solution is depen-dent on the order of polynomials considered, and therefore on the truncationin the polynomial chaos expansion. Also, convergence requires the solutionto be smooth with respect to the parameters describing the input uncertainty[63].
1.2 OutlineThe aim of Chapter 2 and Chapter 3 of this thesis is to give a theoretical back-ground for the numerical and theoretical results to be presented in subsequentchapters. The theory of spectral expansions of random fields is outlined inChapter 2, followed by an exposition of methods for the solution of partialdifferential equations with stochastic input in Chapter 3. Numerical discretiza-tion schemes are described in Chapter 4.
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2. Random field representation
Nonlinear conservation laws subject to uncertainty are expected to developsolutions that are discontinuous in the spatial as well as in the stochastic di-mensions. In order to allow piecewise continuous solutions to the problemsof interest, we follow [21] and broaden the concept of solutions to the class offunctions equivalent to a function f , denoted C f , and define a normed spacethat does not require its elements to be smooth functions. Let (Ω,A ,P) be aprobability space with event space Ω, and probability measure P defined onthe σ -field A of subsets of Ω. Let ξξξ = ξ j(ω)N
j=1 be a set of N indepen-dent and identically distributed random variables for ω ∈ Ω. We consider ubelonging to the space
L2(Ω,P) =
C f | f measurable w.r.t.P;
∫Ω
f 2dP(ξ )< ∞
. (2.1)
The inner product between two functionals a(ξ ) and b(ξ ) belonging to L2(Ω,P)is defined by
〈a(ξ )b(ξ )〉=∫
Ω
a(ξ )b(ξ )dP(ξ ). (2.2)
This inner product induces the norm ‖ f‖2L2(Ω,P) = 〈 f 2〉.
Throughout the thesis, we will require that all random quantities are second-order random fields, i.e. any f (ξ ) is subject to the constraint
‖ f‖2L2(Ω,P) =
∫Ω
f 2dP(ξ )< ∞.
Spectral representations of random functionals aims at finding a series expan-sion in the form
f (ω) =∞
∑k=0
fkψk(ξ (ω)),
where ψk(ξ )∞k=0 is the set of basis functions and fk∞
k=0 is the set of coef-ficients to be determined.
2.1 Karhunen-Loève expansionStarting with a set of data, the Karhunen-Loève expansion [31, 39], also knownas proper orthogonal decomposition or principal component analysis, provides
11
a series representation of a random field in terms of its spatial correlation (co-variance kernel). Any second-order random field f (x,ω) can be representedby the Karhunen-Loève expansion,
f (x,ω) = f (x)+∞
∑k=1
ηk(ω)√
λkφk(x),
where f (x) is the mean of f (x,ω), the random variables ηk are uncorrelatedwith mean zero, and λk and φk are the eigenvalues and eigenfunctions of thecovariance kernel, respectively.
For random fields with known covariance structure, the Karhunen-Loèveexpansion is optimal in the sense of minimizing the mean-square error. Thecovariance function of the output of a problem is in general not known a pri-ori. However, Karhunen-Loève representations of the input data can often becombined with generalized chaos expansions, to be presented in the next sec-tion. The principal components are identified and the random field is thenapproximated by generalized chaos expansion in each random component ηk,for k = 1, . . . , p, for some order p.
2.2 Generalized chaos expansions2.2.1 Generalized polynomial chaos expansionThe polynomial chaos (PC) framework based on series expansions of Her-mite polynomials of Gaussian random variables was introduced by Ghanemand Spanos [27] and builds on the theory of homogeneous chaos introducedby Wiener in 1938 [65]. Any second order random field can be expanded as ageneralized Fourier series in the set of orthogonal Hermite polynomials, whichconstitutes a complete basis in the Hilbert space L2(Ω,P) defined by (2.1).The resulting polynomial chaos series converges in the L2(Ω,P) sense as aconsequence of the Cameron-Martin theorem [10]. Although not limited torepresent functions with Gaussian distribution, the polynomial chaos expan-sion achieves the highest convergence rate for Gaussian functions. Xiu andKarniadakis [72] generalized the polynomial chaos framework to the gener-alized polynomial chaos (gPC) expansion, where random functions are rep-resented by any set of hypergeometric polynomials from the Askey scheme[5]. A function with uniform distribution is optimally represented by Legen-dre polynomials that are orthogonal with respect to the uniform measure, anda gamma distributed input by Laguerre polynomials that are orthogonal withrespect to the gamma measure etc. The optimality of the choice of stochasticexpansion pertains to the representation of the input; the representation of theoutput of a nonlinear problem may well be highly nonlinear as expressed inthe basis of the input.
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The Cameron-Martin theorem applies also to gPC with non-Gaussian ran-dom variables, but only when the probability measure P(ξ ) of the stochasticexpansion variable ξ is uniquely determined by the sequence of moments,
〈ξ k〉=∫
Ω
ξkdP(ξ ), k ∈ N0, (2.3)
where N0 denotes the non-negative integers. Property (2.3) is not always satis-fied in situations commonly encountered; for instance, the lognormal general-ized chaos does not satisfy this property. Thus, there are cases when the gPCexpansion does not converge to the true limit of the random variable underexpansion [20]. However, lognormal random variables may be successfullyrepresented by gPC satisfying the determinacy of moments, e.g. Hermite poly-nomial chaos expansion. This motivates our choice to use Hermite polynomialchaos expansion to represent lognormal viscosity in Paper III.
Consider a generalized chaos basis ψi(ξ )∞i=0 spanning the space of sec-
ond order (i.e. finite variance) random processes on this probability space.The basis functionals are assumed to be orthonormal, i.e. they satisfy
〈ψiψ j〉= δi j. (2.4)
Any second order random field u(x, t,ξ ) can be expressed as
u(x, t,ξ ) =∞
∑i=0
ui(x, t)ψi(ξ ), (2.5)
where the coefficients ui(x, t) are defined by the projections
ui(x, t) = 〈u(x, t,ξ )ψi(ξ )〉, i = 0,1, .... (2.6)
Independent of the choice of basis ψi∞i=0, we can express the mean and
variance of u(x, t,ξ ) as
E(u(x, t,ξ )) =∫
Ω
u(x, t,ξ )dP(ξ ) = u0(x, t),
and
Var(u(x, t,ξ )) =∫
Ω
(u(x, t,ξ )−E(u(x, t,ξ ))2dP(ξ ) =∞
∑i=1
u2i (x, t),
respectively. For practical purposes, (2.5) is truncated to a finite number Mterms, and we set
u(x, t,ξ )≈ uM(x, t,ξ ) =M
∑i=0
ui(x, t)ψi(ξ ). (2.7)
The number of basis functions M+1 is dependent on the number of stochas-tic dimensions N and the order of truncation of the generalized chaos expan-sion. Assuming the same order p of generalized chaos expansion for each
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stochastic dimension, the relation M+1= (N+ p)!/(N!p!) holds. Thus, high-dimensional problems tend to become computationally infeasible and mucheffort has been put into alleviating the computational cost through e.g. sparse-grid representations [52, 24]. In this thesis, only one stochastic dimension willbe considered, and the issue of the so-called curse of dimensionality will notbe further addressed.
The basis ψi∞i=0 is often a set of orthogonal polynomials. Given the two
lowest order polynomials, higher order polynomials can be generated by therecurrence relation
ψn(ξ ) = (anξ +bn)ψn−1(ξ )+ cnψn−2(ξ ),
where the coefficients an, bn, cn are specific to the class of polynomials.The truncated chaos series (2.7) may result in solutions that are unphysi-
cal. An extreme example is when a strictly positive quantity, say density, withuncertainty within a bounded range is represented by a polynomial expansionwith infinite range, for instance Hermite polynomials of standard Gaussianvariables. The Hermite series expansion converges to the true density withbounded range in the limit M → ∞, but for a given order of expansion, sayM = 1, the representation ρ = ρ0 +ρ1H1(ξ ) results in negative density withnon-zero probability since the Hermite polynomial H1 takes arbitrarily largenegative values. Similar problems may be encountered also for polynomialrepresentations with bounded support. Polynomial reconstruction of a dis-continuity in stochastic space leads to Gibbs oscillations that may yield nega-tive values of an approximation of a solution that is close to zero but strictlypositive by definition. Whenever discontinuities are involved, one should becareful with the use of global polynomial representations.
Spectral convergence of the generalized polynomial chaos expansion is ob-served when the solutions are sufficiently regular and continuous [72], but forgeneral non-linear conservation laws - such as in fluid dynamics problems -the convergence is usually less favorable. Spectral expansion representationsare still of interest for these problems because of the potential efficiency withrespect to brute force sampling methods, but special attention must be devotedto the numerical methodology used. For some problems with steep gradientsin the stochastic dimensions, polynomial chaos expansions fail entirely to cap-ture the solution [36]. Global methods can still give superior overall perfor-mance, for instance Padé approximation methods based on rational functionapproximation [14], and hierarchical wavelet methods that are global methodswith localized support of each resolution level [34]. These methods do notneed input such as mesh refinement parameters and they are not dependent onthe initial discretization of the stochastic space.
An alternative to polynomial expansions for non-smooth and oscillatoryproblems is generalized chaos based on a localization or discretization of thestochastic space [17, 46]. Methods based on stochastic discretization suchas adaptive stochastic multi-elements [62] and stochastic simplex collocation
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[67] will be briefly described in section 3.2.3. The robust properties of dis-cretized stochastic space can also be obtained by globally defined wavelets,see [34, 35]. The next section outlines piecewise linear Haar wavelet chaos,followed by an exposition about piecewise polynomial multiwavelet general-ized chaos. These classes of basis functions are robust to discontinuities. Weuse Haar wavelets and other multiwavelets in Paper IV and V.
2.2.2 Haar wavelet expansionHaar wavelets are defined hierarchically on different resolution levels, repre-senting successively finer features of the solution with increasing resolution.They have non-overlapping support within each resolution level, and in thissense they are localized. Still, the Haar basis is global due to the overlappingsupport of wavelets belonging to different resolution levels. Haar waveletsdo not exhibit spectral convergence, but reduce the Gibbs phenomenon in theproximity of discontinuities in the stochastic dimension.
Consider the mother wavelet function defined by
ψW (y) =
1 0≤ y < 12
−1 12 ≤ y < 1
0 otherwise,(2.8)
Based on (2.8) we get the wavelet family
ψWj,k(y) = 2 j/2
ψW (2 jy− k), j = 0,1, ...; k = 0, ...,2 j−1,
Given the probability measure of the stochastic variable ξ with cumulativedistribution function Fξ (ξ0) = P(ω : ξ (ω)≤ ξ0), define the basis functions
Wj,k(ξ ) = ψWj,k(Fξ (ξ ))
Adding the basis function W0(y) = 1 in y∈ [0,1] and concatenating the indicesj and k into i = 2 j + k so that Wi(ξ ) ≡ ψW
n,k(Fξ (ξ )), we can represent anyrandom variable u(x, t,ξ ) with finite variance as
u(x, t,ξ ) =∞
∑i=0
ui(x, t)Wi(ξ ),
which is of the form (2.5). Figure 2.1 depicts the first eight basis functions ofthe generalized Haar wavelet chaos.
15
−1 −0.5 0 0.5 1
−2
−1
0
1
2
Resoluti on l eve l 0: W 0
−1 −0.5 0 0.5 1
−2
−1
0
1
2
Resoluti on l eve l 0: W 1
−1 −0.5 0 0.5 1
−2
−1
0
1
2
Resoluti on l eve l 1: W 2, W 3
−1 −0.5 0 0.5 1
−2
−1
0
1
2
Resoluti on l eve l 2: W 4, W 5, W 6, W 7
Figure 2.1. Haar wavelets, resolution levels 0,1,2.
2.2.3 Multiwavelet expansionThe main idea of multiwavelets is to combine the localized and hierarchi-cal structure of Haar wavelets with the convergence properties of orthogonalpolynomials by complementing the piecewise constant wavelets with piece-wise polynomial wavelets. The procedure of constructing these multiwaveletsusing Legendre polynomials follows the algorithm in [4, 35].
An alternative to gPC expansions for non-smooth and oscillatory problemsis generalized chaos based on a localization or discretization of the stochasticspace [17, 46]. Methods based on stochastic discretization include adaptivestochastic multi-elements [62] and stochastic simplex collocation [67]. Therobust properties of discretized stochastic space can also be obtained by glob-ally defined wavelets, see [34]. In this paper, we follow the approach of [35]and use piecewise polynomial multiwavelets (MW), defined on sub-intervalsof [−1,1]. The construction of a truncated MW basis follows the algorithm in[4].
Starting with the space VNp of polynomials of degree at most Np defined onthe interval [−1,1], the construction of multiwavelets aims at finding a basisof piecewise polynomials for the orthogonal complement of VNp in the spaceVNp+1 of polynomials of degree at most Np+1. Merging the bases of VNp andthat of the orthogonal complement of VNp in VNp+1, we obtain a piecewisepolynomial basis for VNp+1. Continuing the process of finding orthogonalcomplements in spaces of increasing degree of piecewise polynomials, leadsto a basis for L2([−1,1]).
We first introduce a smooth polynomial basis on [−1,1]. Let Lei(ξ )∞i=0
be the set of Legendre polynomials that are defined on [−1,1] and orthogonal
16
with respect to the uniform measure. The normalized Legendre polynomialsare defined recursively by
Le j+1(ξ ) =√
2 j+3(√
2 j+1j+1
ξ Le j(ξ )−j
( j+1)√
2 j−1Le j−1(ξ )
),
Le0(ξ ) = 1, Le1(ξ ) =√
3ξ .
The set Lei(ξ )Npi=0 is an orthonormal basis for VNp . Double products are
readily computed from (2.4), and higher-order products are precomputed usingnumerical integration.
Following the algorithm by Alpert [4] (see Appendix of Paper IV), we con-struct a set of mother wavelets ψW
i (ξ )Npi=0 defined on the domain ξ ∈ [−1,1],
where
ψWi (ξ ) =
pi(ξ ) −1≤ ξ < 0(−1)Np+i+1 pi(ξ ) 0≤ ξ < 10 otherwise,
(2.9)
where pi(ξ ) is an ith order polynomial. By construction, the set of waveletsψW
i (ξ )Npi=0 are orthogonal to all polynomials of order at most Np, hence the
wavelets are orthogonal to the set Lei(ξ )Npi=0 of Legendre polynomials of
order at most Np. Based on translations and dilations of (2.9), we get thewavelet family
ψWi, j,k(ξ )= 2 j/2
ψWi (2 j
ξ−k), i= 0, ...,Np, j = 0,1, ..., k= 0, ...,2 j−1.
Let ψm(ξ ) for m = 0, ...,Np be the set of Legendre polynomials up to orderNp, and concatenate the indices i, j,k into m = (Np + 1)(2 j + k− 1) + i sothat ψm(ξ ) ≡ ψW
i, j,k(ξ ) for m > Np. With the MW basis ψm(ξ )∞m=0 we can
represent any random variable u(x, t,ξ ) with finite variance as
u(x, t,ξ ) =∞
∑m=0
um(x, t)ψm(ξ ),
which is of the form (2.5). In the computations, we truncate the MW seriesboth in terms of the piecewise polynomial order Np and the resolution levelNr. With the index j = 0, ...,Nr, we retain M = (Np +1)2Nr terms of the MWexpansion.
The truncated MW basis is characterized by the piecewise polynomial orderNp and the number of resolution levels Nr, illustrated in Figure 2.2 for Np = 2and Nr = 3. As special cases of the MW basis, we obtain the Legendre poly-nomial basis for Nr = 0 (i = j = 0), and the Haar wavelet basis of piecewiseconstant functions for Np = 0.
17
−1 −0.5 0 0.5 1−2
−1
0
1
2
3
Legen d r e p o l yn om i a l s
−1 −0.5 0 0.5 1−4
−2
0
2
4
Reso l u t i on l evel 0 , o r t h . comp l emen t
−1 −0.5 0 0.5 1−5
0
5
Reso l u t i on l evel 1
−1 −0.5 0 0.5 1−10
−5
0
5
10
Reso l u t i on l evel 2
Figure 2.2. Multiwavelets for Np = 2, Nr = 3. Resolution level 0 consists of the firstNp +1 Legendre polynomials and their orthogonal complement. Resolution level j >0 contains (Np + 1)2 j wavelets each. Each basis function is a piecewise polynomialof order Np.
18
3. Polynomial chaos methods
In this chapter we review methods for formulating partial differential equa-tions based on the random field representations outlined in Chapter 2. Theseinclude the stochastic Galerkin method, which is the predominant choice inthis thesis, as well as other methods that frequently occur in the literature. Wealso include methods that are not polynomial chaos methods themselves butviable alternatives to these.
3.1 Intrusive methods3.1.1 Stochastic Galerkin methodsThe stochastic Galerkin method was introduced by Ghanem and Spanos inorder to solve linear stochastic equations [27]. It relies on a weak problemformulation where the set of solution basis functions (trial functions) is thesame as the space of stochastic test functions. Consider a general conservationlaw defined on a spatial domain Ωx with boundary Γx subject to initial andboundary conditions, given by
∂u(x, t,ξ )∂ t
+∂ f (u(x, t,ξ ),ξ )
∂x= 0, x ∈Ωx, t ≥ 0, (3.1)
L (u,x, t,ξ ) = g(t,ξ ), x ∈ Γx, t ≥ 0, (3.2)u = h(x,ξ ), x ∈Ωx, t = 0. (3.3)
where u is the solution and f is the flux function. A weak approximation of3.1 is obtained by substituting the truncated gPC series of the solution u givenby (2.7) into (3.1) and projecting the resulting expression onto the subspaceof L2(Ω,P) spanned by the truncated basis ψi(ξ )M
i=0. The result is thestochastic Galerkin formulation of (3.1),
∂uk(x, t)∂ t
+∂
∂x
⟨f
(M
∑i=0
uiψi(ξ ),ξ
),ψk
⟩=0, x ∈Ωx, t ≥ 0, (3.4)
〈L (u,x, t,ξ ),ψk〉=〈g,ψk〉 , x ∈ Γx, t ≥ 0, (3.5)〈u,ψk〉=〈h,ψk〉 , x ∈Ωx, t = 0. (3.6)
for k = 0, ...,M, where the inner product 〈., .〉 is defined in (2.2). Althoughprevalent in the literature, there are situations, even for linear problems, when
19
it is essential not to restrict the gPC approximations of all input quantities (e.g.material parameters) to the same order M as the gPC representation of the so-lution. An example is given in Paper III, where we show that the stochasticGalerkin formulation of an advection-diffusion equation may lead to an ill-posed problem unless an order at least 2M approximation of the diffusion pa-rameter is used whenever an order M gPC approximation is used to representthe solution.
The stochastic Galerkin formulation (3.4) is an extended deterministic sys-tem of coupled equations. In general, it is different from the correspondingdeterministic problem and therefore needs to be solved using different numer-ical solvers. Compared to sample-based methods where a number of decou-pled equations are solved, the number of coupled equations of the stochasticGalerkin method may be significantly smaller. This is true in particular formulti-dimensional problems [73].
Intrusive generalized chaos methods for nonlinear conservation laws havebeen investigated in e.g. [59], where a reduced-cost Roe solver with entropycorrector was presented, and in [58] with different localized representationsof uncertainty in initial functions and problem coefficients. Poëtte et. al. [47]used a nonlinear projection method to bound the oscillations close to stochas-tic discontinuities by polynomial chaos expansion of the entropy variables ob-tained from a transformation of the conservative variables.
3.1.2 Semi-intrusive methodsAlternative approaches to generalized chaos methods have also been presentedin the literature. Abgrall et. al. [2, 3] developed a semi-intrusive method basedon a finite-volume like reconstruction technique of the discretized stochasticspace. A deterministic problem is obtained by taking conditional expectationsgiven a stochastic subcell, over which ENO constructions are used to recon-struct the fluxes in the stochastic dimensions. This makes it particularly suit-able for non-smooth probability distributions, in contrast to gPC, where theconvergence requires the solution to be smooth with respect to the parametersdescribing the input uncertainty [63].
3.2 Non-intrusive methodsAn alternative to the polynomial chaos approach with stochastic Galerkin pro-jection is to use multiple samples of solutions corresponding to some real-izations of the stochastic inputs. Such non-intrusive methods do not requiremodification of existing codes but rely exclusively on repeated runs of thedeterministic code which make them computationally attractive.
20
3.2.1 Stochastic collocation methodsAn alternative to gPC methods is the class of stochastic collocation methods,i.e., sampling methods with interpolation in the stochastic space, c.f. [71].Several investigations of the relative performance of stochastic Galerkin andcollocation methods have been performed, c.f. [41, 7, 60]. The significant sizeof the stochastic Galerkin system may lead to inefficient direct implementa-tions compared to collocation methods and preconditioned iterative Krylovsubspace methods. However, the use of suitable techniques for large systems,such as preconditioners, may result in speedup for the solution of stochas-tic Galerkin systems compared to multiple collocation runs [60]. For high-dimensional problems where the collocation methods tend to become pro-hibitively expensive, sparse grid adaptive methods have been suggested to al-leviate the computational cost [22].
Stochastic collocation takes a set of solutions u( j) evaluated at a set ξ ( j)of values of random input ξ , and constructs an interpolating polynomial fromthese solution realizations [40, 71, 6]. A common choice of interpolation poly-nomials is the set of Lagrange polynomials L (Mint)
j (ξ )Mintj=1, defined by Mint
points ξ ( j)Mintj=1, for which the polynomial interpolant becomes
I u =Mint
∑j=1
u( j)L j(ξ ). (3.7)
The distribution of the grid points ξ ( j)Mintj=1 is implied by the measure P of
ξ . For instance, we choose ξ ( j) to be the set of Gauss-Legendre quadraturepoints for the case of uniformly distributed µ , and the set of Gauss-Hermitequadrature points for the case of lognormal µ . The integral statistics of inter-est, such as moments, may then be approximated by the corresponding quadra-ture rules. For instance, for some quantity of interest 〈S(u)〉, we have
〈S(u)〉 ≈Mint
∑j=1
S(u( j))w j, (3.8)
where w j is the weight corresponding to the quadrature point ξ ( j). The quadra-ture points and weights can be computed through the Golub-Welsch algorithm[28]. Note that there is no need to find the Lagrange polynomials of (3.7)explicitly since (I u)(ξ ( j)) = u( j) and we only need the values of I u at thequadrature points in (3.8).
Stochastic collocation is similar to other non-intrusive methods such aspseudospectral projection [49] and stochastic point collocation (stochastic re-sponse surfaces) [9], in that it relies on evaluating deterministic solutions asso-ciated with stochastic quadrature points. The difference is the postprocessingstep where quantities of interest are reconstructed by different means of nu-merical quadrature. Specifically, in stochastic collocation, quantities of inter-
21
est are computed directly without representing the solutions as a gPC series.Pseudospectral projection, on the other hand, involves the computation of thepolynomial chaos coefficients of u through numerical quadrature. Quantitiesof interest are then calculated as functions of the polynomial chaos coeffi-cients.
We note that numerical properties of the solutions of the PDE’s, e.g. mono-tonicity, stiffness and stability, depend only on the numerical solution valuesat the quadrature points. Thus, these properties are independent of the post-processing step in which the solution statistics are reconstructed. Therefore,when we compare stochastic Galerkin and stochastic collocation in Paper III,the comparison should be valid for a larger class of non-intrusive methods thanstochastic interpolation.
3.2.2 Spectral projectionSpectral projection, discrete projection or the pseudo-spectral approach [49,69] comprise a set of gPC based methods relying on deterministic solutionsevaluated at sampling points of the parameter domain. They are sometimesreferred to as a subgroup of the class of collocation methods [70]. Alternativespectral projection approaches include weighted least squares formulations fordetermining the gPC coefficients (2.6) [30].
The integrals over the stochastic domain of the gPC projections definedby (2.6) are approximated by sampling or employing numerical quadrature.For multiple stochastic dimensions, sparse grids are attractive, e.g. Smolyakquadrature [32].
3.2.3 Stochastic multi-elementsIn multi-element generalized polynomial chaos (ME-gPC), the stochastic do-main is decomposed into subdomains, and generalized polynomial chaos isapplied element-wise [62, 64]. Local orthogonal polynomial bases can be con-structed numerically using the Stieltjes procedure or the modified Chebyshevalgorithm [23]. The stochastic Galerkin method may be applied element-wiseand in this sense the ME-gPC is an intrusive method.
The multi-element framework allows for the combination of refinement ofthe number of elements (h-refinement) and increasing the order gPC of eachelement (p-refinement) [62]. Variants of these methods that have been ap-plied to hyperbolic problems include element-wise interpolation of stochasticLagrange polynomials applied to a transonic airfoil problem [66], and proba-bilistic collocation applied to a problem of supersonic flow past a wedge [38].
22
4. Spatial discretization
The problems investigated in this thesis can all be written as one-dimensionalconservation laws,
ut + f (u)x = 0, 0≤ x≤ 1, t ≥ 0, (4.1)
where u is the solution vector, and f is a flux function.When solving (4.1) on a uniform grid, we will use two different classes
of numerical schemes. For smooth problems, we use high-order finite differ-ence schemes, and for non-smooth problems we apply shock capturing finitevolume methods.
Summation by parts (SBP) is the discrete equivalent to integration by parts.SBP operators are used for approximations of spatial derivatives. Their use-fulness lies in the possibility of expressing energy decay in terms of knownboundary values, exactly as in the continuous case [33, 53, 43]. For smoothproblems, one can often prove that the numerical methods are stable and high-order accurate.
As a consequence of the formal high-order accuracy of SBP operators, so-lutions with multiple discontinuities are not well-captured. Instead, a morerobust and accurate method such as the MUSCL scheme [61] or the HLL Rie-mann solver [29] with flux limiting, to be described in Section 4.2, will beused.
4.1 Summation-by-parts operatorsIn order to obtain stability of the semi-discretized problem for various ordersof accuracy and non-periodic boundary conditions, we use discrete operatorssatisfying a summation-by-parts (SBP) property [33].
Boundary conditions are imposed weakly through penalty terms, where thepenalty parameters are chosen such that the numerical method is stable. Thisis done through a Simultaneous Approximation Term (SAT), introduced in[11]. Operators of order 2n, n ∈ N, in the interior of the domain are combinedwith boundary closures of order of accuracy n. For the advection-diffusionequation, this leads to the global order of accuracy min(n+2,2n). We refer to[54] for a derivation of this result on accuracy.
The first and second derivative SBP operator were introduced in [33, 53]and [12, 42], respectively. For the first derivative, we use the approximationux ≈ P−1Qu, where subscript x denotes partial derivative and Q satisfies
Q+QT = diag(−1,0, . . . ,0,1)≡ B. (4.2)
23
Additionally, the matrix P is symmetric and positive definite in order todefine a discrete norm.
For the approximation of the second derivative, we can either use the firstderivative operator twice, or use uxx ≈ P−1(−M+ BD)u, where M+MT ≥ 0,B is given by (4.2), and D is a first-derivative approximation at the boundaries,i.e.,
D =1
∆x
d1 d2 d3 . . .
1. . .
1. . . −d3 −d2 −d1
,where di, i= 1,2,3, . . . , are scalar values leading to a consistent first-derivativeapproximation at the boundaries [12, 42].
4.1.1 Artificial dissipation operatorsAn artificial dissipation operator is a scaled discretized even order derivativewhich is added to the system to allow for stable and accurate solutions to beobtained in the presence of solution discontinuities. The artificial dissipation isdesigned to transform the central discretization into a one-sided operator closeto the shock location. Depending on the accuracy of the difference scheme,this requires one or more dissipation operators. All dissipation operators usedhere are of the form
A2k =−∆xP−1DTk BwDk, (4.3)
where P−1 is the diagonal norm of the first derivative as before, D is an ap-proximation of (∆x)k∂ k/∂xk and Bw is a diagonal positive definite matrix tobe chosen within the order of accuracy of the scheme. In most cases here,Bw is replaced by a single dissipation constant βw. An appropriate choice ofdissipation constant results in an upwind scheme, suitable for problems whereshocks evolve. The scaling with P−1 in (4.3) guarantees that an energy-stablescheme remains energy-stable after adding artificial dissipation. For furtherreading about the design of artificial dissipation operators we refer to [43].
4.1.2 Multiple spatial domains with weak SBP-SAT couplingDepending on the local smoothness properties of a problem, it may be at-tractive to use locally adapted difference schemes. The SBP framework canbe applied to multiple domains that are weakly coupled through a numericalinterface. The interface condition is satisfied in a manner similar to the impo-sition of boundary conditions through penalty matrices. In this way, one cancombine schemes of different orders of accuracy. For many problems, one canshow stability and conservation at the interface [12, 13].
24
The MUSCL scheme, that is specifically designed to capture solution dis-continuities, can be rewritten in SBP operator form with an artificial dissi-pation term [1] and can therefore be coupled with other schemes using SBPoperators [19]. In Paper V, this property is used in deriving stable methodsfor a two-phase problem where we apply a variant of the MUSCL scheme tonon-smooth solution regions, and high order SBP operators to smooth solutionregions.
4.2 Shock capturing methodsFor finite volume methods on structured grids we partition the computationaldomain into cells of equal size ∆x. Solution values ui are defined as cell aver-ages of cell i, and fluxes are defined on the edges of the cells.
4.2.1 MUSCL schemeThe MUSCL (Monotone Upstream-centered Schemes for Conservation Laws)scheme was introduced in [61]. Let m be the number of spatial points and ∆x=1/(m− 1) and let u be the spatial discretization of u. The semi-discretizedform of (4.1) is given by
du j
dt+
Fj+1/2−Fj−1/2
∆x= 0, j = 1, ...,m, (4.4)
where Fj+1/2 denotes the numerical flux function evaluated at the interfacebetween cells j and j+1.
For the MUSCL scheme with slope limited states uL and uR, we take thenumerical flux function at the interface between cell j and cell j+1 to be
Fj+ 12=
12
(f (uL
j+ 12)+ f (uR
j+ 12))+
12|J j+ 1
2|(
uLj+ 1
2−uR
j+ 12
), (4.5)
where J is an approximation of the flux Jacobian J = ∂ f/∂u, from which isderived the absolute value |J j+ 1
2| given by
|J j+ 12|= X
∣∣∣Λ(u j+ 12)∣∣∣X−1 =
12
X∣∣∣Λ(uL
j+ 12)+Λ(uR
j+ 12)∣∣∣X−1, (4.6)
where Λ is a diagonal matrix with the eigenvalues of J and X is the eigenvectormatrix. J can be an average of the true Jacobian evaluated at the discretizationpoints or a Roe average matrix [51].
The left (L) and right (R) solution states are given by, respectively
uLj+ 1
2= u j +0.5φ(r j)(u j+1−u j),
uRj+ 1
2= u j+1−0.5φ(r j+1)(u j+2−u j+1).
25
The flux limiter φ(r j) takes the argument r j = (u j−u j−1)/(u j+1−u j). As aspecial case, φ = 0 results in the first-order accurate upwind scheme. Secondorder accurate and total variation diminishing schemes are obtained for φ thatare within the region
φ(r) = 0, r ≤ 0,r ≤ φ(r)≤ 2r, 0≤ r ≤ 1,1≤ φ(r)≤ r, 1≤ r ≤ 2,1≤ φ(r)≤ 2, r ≥ 2,φ(1) = 1,
as defined in [55]. The minmod, van Leer and superbee limiters that are usedin this thesis are all second order and total variation diminishing. For a moredetailed description of the MUSCL scheme, see e.g. [37].
4.2.2 HLL Riemann solverAs a simpler alternative to the MUSCL-Roe solver, we use the HLL (afterHarten, Lax and van Leer) Riemann solver introduced in [29] and further de-veloped in [18]. Instead of computing the Roe average matrix needed for theRoe flux (4.5), only the fastest signal velocities need to be estimated for theHLL solver. These signal velocities SL and SR are the estimated maximum andminimum eigenvalues of the Jacobian J = ∂ f/∂u of the flux.
The HLL flux is defined by
Fj+ 12=
f(
uLj+ 1
2
)if SL ≥ 0
SR f(
uLj+ 1
2
)−SL f
(uR
j+ 12
)+SLSR
(uR
j+ 12−uL
j+ 12
)SR−SL
if SL < 0 < SR
f(
uRj+ 1
2
)if SR ≤ 0
.
In general, obtaining accurate eigenvalue estimates may be computationallycostly. However, for certain choices of stochastic basis functions in combina-tion with known eigenvalues of the deterministic system, we derive analyticalexpressions for the stochastic Galerkin system eigenvalues [45].
The HLL-flux approximates the solution by assuming three states separatedby two waves. In the deterministic case, this approximation is known to failin capturing contact discontinuities and material interfaces of solutions to sys-tems with more than two waves [56]. For the Euler equations, the contactsurface can be restored by using the HLLC solver where three waves are as-sumed [57]. The stochastic Galerkin system is a multi-wave generalization ofthe deterministic case, and similar problems in capturing missing waves areexpected. However, the robustness and simplicity of the HLL-solver makes it
26
a potentially more suitable choice compared to other Riemann solvers that aretheoretically more accurate, but also more sensitive to ill-conditioning of thesystem matrix.
The HLL solver applied directly to the variable vector results in excessivenumerical diffusion [56], but by applying flux limiters in the same way asin the MUSCL scheme for higher-order reconstruction, sharp features of thesolution are recovered.
27
5. Summary of papers
5.1 Paper IIn this paper we study the inviscid Burgers’ equation subject to uncertaintyin the initial and boundary conditions. We use a polynomial chaos (PC) ex-pansion approach where the solution is represented as a truncated series ofstochastic, orthogonal polynomials. The analysis of well-posedness for thesystem resulting after Galerkin projection is presented and follows the pat-tern of the corresponding deterministic Burgers’ equation. The numerical dis-cretization is based on spatial derivative operators satisfying the summation byparts property and weak boundary conditions are enforced to ensure stability.Similarly to the deterministic case, the explicit time step for the hyperbolicstochastic problem is proportional to the inverse of the largest eigenvalue ofthe system matrix. The time step naturally decreases compared to the deter-ministic case since the spectral radius of the continuous problem grows withthe number of polynomial chaos coefficients. An estimate of the eigenvalues isprovided. A characteristic analysis of the truncated PC system is presented andgives a qualitative description of the development of the system over time fordifferent initial and boundary conditions. It is shown that a precise statisticalcharacterization of the input uncertainty is required and partial information,e.g. the expected values and the variance, are not sufficient to obtain a solu-tion. An analytical solution is derived and the coefficients of the infinite PCexpansion are shown to be smooth, while the corresponding coefficients of thetruncated expansion are shown to be discontinuous.
5.2 Paper IIWe continue the work in Paper I by considering the effect of lack of boundarydata for the stochastic Galerkin formulation of Burgers’ equation. A first orderpolynomial stochastic Galerkin approximation with known analytical solutionis considered. Different settings and techniques for imposition of unknownboundary data are investigated. Extrapolation of boundary data from the inte-rior of the computational domain leads to an accurate numerical solution butnot to stability using the energy method. Enforcing the initial boundary dataat all times leads to stability but the numerical solution deviates from the truesolution and converges to a different steady-state solution.
28
5.3 Paper IIIThe stochastic Galerkin and collocation methods are used to solve an advection-diffusion equation with uncertain and spatially varying viscosity. We investi-gate well-posedness, monotonicity and stability for the extended system re-sulting from the Galerkin projection of the advection-diffusion equation ontothe stochastic basis functions. High-order summation-by-parts operators andweak imposition of boundary conditions are used to prove stability of the semi-discrete system.
It is essential that the eigenvalues of the resulting viscosity matrix of thestochastic Galerkin system are positive and we investigate conditions for thisto hold. When the viscosity matrix is diagonalizable, stochastic Galerkin andstochastic collocation are similar in terms of computational cost, and for somecases the accuracy is higher for stochastic Galerkin provided that monotonicityrequirements are met. We also investigate the total spatial operator of the semi-discretized system and its impact on the convergence to steady-state.
For diffusive problems, the stochastic Galerkin formulation leads to betteraccuracy compared to stochastic collocation. For steady-state calculations,stochastic collocation is faster for advection dominated cases and stochasticGalerkin is faster for diffusive cases.
5.4 Paper IVThe Euler equations subject to uncertainty in the input parameters are investi-gated via the stochastic Galerkin approach. We present a new fully intrusivemethod based on a variable transformation of the continuous equations. Roevariables are employed to get quadratic dependence in the flux function and awell-defined Roe average matrix that can be determined without matrix inver-sion.
In previous formulations based on generalized chaos expansion of the phys-ical variables, the need to introduce stochastic expansions of inverse quanti-ties, or square-roots of stochastic quantities of interest, leads to a number ofpossible different ways to approximate the original stochastic problem. Someof these psuedo-spectral operations, such as the approximation of stochasticinverse quantities, are ill-conditioned and should be avoided if possible. Wepresent a method where no auxiliary quantities are needed, resulting in anunambiguous problem formulation.
We use the MUSCL scheme with flux limiters to solve the system in theRoe variables. The Roe flux requires a Roe average matrix with certain shock-capturing properties. We prove the existence of a Roe average matrix for thestochastic Galerkin formulation in Roe variables.
The Roe formulation saves computational cost compared to the formulationbased on expansion of conservative variables. Moreover, the Roe formulationis more robust and can handle cases of supersonic flow, for which the con-
29
servative variable formulation leads to instability. For more extreme cases,where the global Legendre polynomials poorly approximate discontinuities instochastic space, we use the localized Haar wavelet basis.
5.5 Paper VWe combine the techniques developed in the previous papers and apply themto a two-phase flow problem with input uncertainty. Numerical experimentsare carried out assuming uncertainty in the interface location, but the frame-work generalizes to uncertainty with known distribution in other input data.Uncertainty is represented through a truncated multiwavelet expansion.
We assume that the discontinuous features of the solution are restricted tosubdomains in the computational domain and use a high-order method for thesmooth regions coupled weakly through numerical interfaces with a robustshock capturing method for the non-smooth regions.
The discretization of the non-smooth region is based on a generalizationof the HLL flux, and have many properties in common with its deterministiccounterpart. It is simple and robust, and captures the statistics of the shock.The discretization of the smooth region is carried out with high-order finite-difference operators satisfying a summation-by-parts property.
A symmetrization and combination of conservative and non-conservativeformulation leads to a generalized energy estimate for the stochastic Galerkinsystem, just as for the case of the deterministic Euler equations. Under certainsmoothness assumptions, stability at the interfaces can be obtained for thesymmetrized system. The derived penalty matrices are transformed back to theconservative variable formulation that is used in the numerical experiments.
The numerical error at the interface is dominated by the error due to thetruncation of the multiwavelet expansion. As we increase the order of mul-tiwavelet expansion, the number of discountinuities increases, but the magni-tude of each discontinuity decreases.
30
6. Discussion and outlook
Although funded by King Abdullah University of Science and Technology inSaudi Arabia, the work of this thesis has been performed in close connec-tion to the PSAAP (Predictive Science Academic Alliance Program) projectat Stanford University. In a sense, it has been a subproject within the largePSAAP project, which also involves high-performance computing, physicalmodeling in a complex computational fluid dynamics framework and labora-tory experiments with the common goal of solving prediction problems relatedto hypersonic flight. The ultimate goal with the work presented in this thesisis to extend and integrate the methods into a similar large-scale project.
This thesis treats conservation laws subject to a single source of uncertainty.More realistic complex problems should include multiple sources of uncer-tainty, resulting in multidimensional integrals in the case of non-intrusive for-mulations or large, possible sparse, Galerkin systems. The work in Paper IIIis currently expanded in this direction.
A major limitation of many methods for uncertainty quantification is thatthey become prohibitively expensive for multiple sources of uncertainty. Afeasible method for uncertainty quantification should therefore possess goodscaling properties in terms of computational cost versus the number of stochas-tic dimensions. A relevant comparison of the stochastic Galerkin method withother methods for uncertainty quantification should therefore include multi-variate input uncertainty.
In many numerical methods for complex problems, boundary conditions areprescribed based on the physics of the problem. It may not be straightforwardto interpret and extend these physical boundary conditions to the stochasticGalerkin formulation. Coherent procedures for the imposition of boundarydata are necessary for these problems. In particular, this also holds for situa-tions where boundary data are limited due to lack of measurements.
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7. Stokastiska Galerkinmetoder förkonserveringslagar
Matematisk modellering av fysikaliska problem kräver tillgång till kända initial-och randvillkor samt föreskrivna parametervärden. I många tillämpningar ärinte alla növändiga data exakt givna och en realistisk lösningsmetod kräver attman tar hänsyn till den rådande osäkerheten. Stokastiska metoder kan medfördel tillämpas på sådana problem, oavsett om källan till osäkerhet beror påvariabilitet i mätdata, modellosäkerhet eller annan osäkerhet.
I föreliggande avhandling undersöker vi hyperboliska problem och hyper-bolisk-paraboliska problem med initial- och randvillkor samt materialparame-trar som har okända värden, men för vilka vi antar att vi har en stokastiskmodell i form av entydiga och kända sannolikhetsfördelningar. Ekvationernaär vanligt förekommande modellproblem inom strömningsmekanik.
Den stokastiska lösningen representeras i form av en generaliserad Fouri-erserieutveckling i stokastiska basfunktioner, i regel i form av styckvisa poly-nom. I sammanhanget förekommer ofta begreppet generaliserat kaos. Ordetkaos är i det här sammanhanget en kvarleva från Norbert Wieners ursprung-liga terminologi och syftar på de stokastiska basfunktionerna för representa-tionen av lösningen. Genom att tillämpa den stokastiska Galerkinmetoden påden partiella differentialekvationen projiceras den stokastiska lösningen på destokastiska basfunktionerna. Detta resulterar i ett utökat system av ekvationerför kaos-koefficienterna som är oberoende av den stokastiska parametriserin-gen och kan lösas med klassiska numeriska metoder för deterministiska par-tiella differentialekvationer.
För tillräckligt kontinuerliga problem använder vi finita differensmetodersom är konstruerade för att möjliggöra energiuppskattningar genom partiellsummation. För diskontinuerliga problem använder vi istället mer robustametoder som är anpassade för att approximera lösningar med diskontinuiteter.De senare har låg formell noggrannhetsordning men fångar i allmänhet diskon-tinuiterena bättre än högre ordningens metoder.
Vi presenterar en numerisk metod för att lösa en stokastisk Galerkinformu-lering av Burgers ekvation med stokastiska rand- och initialvillkor. Vi härlederen analytisk stokastisk lösning, analyserar regularitet och visar att kaoskoef-ficienterna är kontinuerliga. Dock är kaoskoefficienterna hos det trunkeradestokastiska Galerkinsystemet diskontinuerliga.
För Burgers ekvation i stokastisk Galerkinform undersöker vi också effek-ten av bristfälliga randdata där vi saknar en komplett sannolikhetsfördelning
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för ränderna. Olika sätt att hantera avsaknad av väntevärde eller varians under-söks numeriskt över tiden genom att extrapolera eller ansätta initiala randdataför alla tider och sedan jämföra den numeriska lösningen med den analytiskalösningen. Extrapolerade randdata resulterar i en formulering för vilken en-ergimetoden inte leder till stabilitet, men som ger en numerisk lösning somöverensstämmer väl med den analytiska lösningen. Konstanta randdata ledertill en stabil numerisk metod, men den numeriska lösningen avviker sigifikantfrån den analytiska lösningen.
Därefter undersöker vi advektion-diffusionsekvationen med stokastisk vis-kositet och jämför den stokastiska Galerkinmetoden med interpolationsme-toder där den deterministiska lösningen beräknas i ett antal stokastiska interpo-lationspunkter. Rättställdhet, monotonicitet, stabilitet och konvergens till denstationära lösningen undersöks analytiskt och numeriskt. Vi visar att valet avrepresentation av den stokastiska viskositeten är avgörande för det stokastiskaGalerkin-systemets numeriska egenskaper. En olämplig representation kanleda till att problemet inte är rättställt.
Vi introducerar också en ny stokastisk Galerkinformulering för Eulers ek-vationer baserad på en variabeltransformation. Genom valet av variabler min-skar vi antalet beräkningar av projektioner av icke-linjära funktioner på destokastiska basfunktionerna. Den här typen av projektioner bör om möjligtundvikas då de är beräkningsmässigt kostsamma och i vissa fall illa-kondition-erade. Flödesfunktionens Jacobian är linjär i de nya lösningsvariablerna, vilketminskar antalet operationer i den s.k. MUSCL-metoden. Vi bevisar att denstokastiska Galerkinformuleringen av Jacobianen av flödesfunktionen besit-ter de egenskaper som definierar Roes medelvärdesmatris och behövs för attdiskontinuiteter i lösningen ska hanteras på rätt sätt. Resultatet av variabel-transformationen är en robust och beräkningseffektiv metod som fungerar förfall av supersoniska flöden då den stokastiska Galerkinmetoden baserad påkonservativa variabler bryter samman.
Slutligen kombinerar och utvecklar vi tidigare resultat i en hybridmetodsom kombinerar partiella-summations-operatorer för kontinuerliga lösningarmed gradientbegränsande metoder för diskontinuerliga lösningar. Vi studerarett stokastiskt tvåfasflödesproblem där vi vet att diskontinuiteterna är begrän-sade till en viss del av beräkningsrummet.
Där de kontinuerliga och diskontinuerliga beräkningsområdena möts finnsen numerisk gränsyta där de olika lösningarna påverkar varandra. Vi visarhur vissa parametervärden skall väljas så att lösningen vid gränsytan är nu-meriskt stabil och bevaras. För de stokastiska representationer vi testar ärdet numeriska felet vid gränsytorna försumbart jämfört med det stokastiskatrunkeringsfelet. Vi visar numeriskt att lösningen konvergerar med antaletstokastiska basfunktioner via ett växande antal diskontinuiteter av minskandemagnitud till en kontinuerlig lösning.
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8. Acknowledgements
First and foremost I would like to thank my advisors Professor Jan Nordströmand Professor Gianluca Iaccarino for their invaluable support during my PhD.I have benefited a lot from their very different and complementing advice andresearch styles. In particular, I have appreciated the encouraging and inspiringatmosphere and many opportunities provided by Professor Iaccarino as wellas Professor Nordström’s commitment to his students and for always findingtime to give feedback. I am also very thankful for their courage to accept meas a doctoral student despite my lack of background in applied mathematics.
I would like to express my gratitude to Professor Alireza Doostan for thecollaboration on the paper about the stochastic advection-diffusion equation.
I am grateful to Professor Rémi Abgrall, Dr. Pietro Congedo and GianlucaGeraci for hosting and welcoming me to INRIA Bordeaux Sud-Ouest and forgiving me valuable feedback on my work.
Thanks to the staff at the Division of Scientific Computing (TDB) at Up-psala University, and in particular my research group, including Dr. QaiserAbbas, Dr. Sofia Eriksson and Dr. Jens Berg. Special thanks to Dr. Anna Nis-sen for proof-reading this thesis and for support and interesting discussionsthroughout the PhD project.
I am grateful for fruitful discussions with my present and former colleaguesat the Center for Turbulence Research and the Uncertainty Quantification groupat Stanford University, in particular Gary Tang, Paul Covington, Dr. DanieleSchiavazzi, Dr. Paul Constantine and Nicolas Kseib.
I would like to thank Dr. Xiangyu Hu and Dr. Kwok Kai So at TU Münchenfor ideas and fruitful discussions regarding numerical methods for two-phaseflow.
I am grateful for the help from the administrators at TDB, Center for Tur-bulence Research and ICME, Carina Lindgren, Marlene Lomuljo-Bautista andIndira Choudhury, respectively.
The funding of this PhD has generously been provided by King AbdullahUniversity of Science and Technology (KAUST) in Saudi Arabia. Financialsupport for the participation in the summer program 2011 at TU München,Germany, was provided by the German Research Foundation (Deutsche For-schungsgemeinschaft - DFG) in the framework of the Sonderforschungsbere-ich Transregio 40 and the IGSSE (International Graduate School of Scienceand Engineering). I am thankful for the resources that have been provided forme during my visits and extended stays at TDB, Uppsala.
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Acta Universitatis UpsaliensisDigital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1008
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A doctoral dissertation from the Faculty of Science andTechnology, Uppsala University, is usually a summary of anumber of papers. A few copies of the complete dissertationare kept at major Swedish research libraries, while thesummary alone is distributed internationally throughthe series Digital Comprehensive Summaries of UppsalaDissertations from the Faculty of Science and Technology.
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