High Order Long-Time Accurate Methods for the Stokes-Darcy System and Uncertainty

89 Pages·2015·0.93 MB·English

Florida State University Libraries
Electronic Theses, Treatises and Dissertations The Graduate School
2015
High Order Long-Time Accurate
Methods for the Stokes-Darcy System and
Uncertainty Quantification of Contaminant
Transport
Dong Sun
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FLORIDA STATE UNIVERSITY
COLLEGE OF ARTS AND SCIENCES
HIGH ORDER LONG-TIME ACCURATE METHODS FOR THE STOKES-DARCY SYSTEM
AND UNCERTAINTY QUANTIFICATION OF CONTAMINANT TRANSPORT
By
DONG SUN
A Dissertation submitted to the
Department of Mathematics
in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
2015
Copyright c 2015 Dong Sun. All Rights Reserved.
⃝
Dong Sun defended this dissertation on 5/12/2015.
The members of the supervisory committee were:
Xiaoming Wang
Professor Co-Directing Dissertation
Max Gunzburger
Professor Co-Directing Dissertation
Xiaoqiang Wang
University Representative
Brian Ewald
Committee Member
Nick Cogan
Committee Member
TheGraduateSchoolhasverifiedandapprovedtheabove-namedcommitteemembers, andcertifies
that the dissertation has been approved in accordance with university requirements.
ii
ACKNOWLEDGMENTS
First I would like to thank my two major professors, Xiaoming Wang and Max Gunzburger. With-
out you two, this dissertation would not be possible and I would not be who I am today. You two
alwaysgaveyourwisdomandinsighttohelpguidemethroughmymorethanfive-year-longjourney
as a Ph.D. student. Wenbin Chen, a senior collaborator, also helped me a lot in both academic and
life experiences. Your enthusiasm about everything is impressive. I also want to thank Xiaoming
He who got me started when I first came here. Daozhi Han and Nan Chen are the other two I want
to thank for their help. Jin Cheng motivated me to choose this path; thank you too. I also want to
thank my family for their ultimate support, especially my grandfather. Thank you all for making
me who I am today.
iii
TABLE OF CONTENTS
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
1 Introduction 1
2 The Stokes–Darcy system and its weak formulation 5
2.1 Conceptual domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Stokes equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 The Darcy equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 The Stokes–Darcy system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.6 Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.7 Saddle point theory and well-posedness of the steady-state problem . . . . . . . . . . 10
3 Two types of second-order IMplicit-EXplicit (IMEX) methods 13
3.1 The second-order backward-differentiation method (BDF2) . . . . . . . . . . . . . . 13
3.1.1 Efficiency of the scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.2 Unconditional stability of the BDF2 scheme . . . . . . . . . . . . . . . . . . . 14
3.1.3 Uniform in time estimates for the BDF2 scheme . . . . . . . . . . . . . . . . 16
3.1.4 Uniform in time H1(Ω) bound of the BDF2 scheme . . . . . . . . . . . . . . 19
3.1.5 Error analysis of the BDF2-FEM scheme . . . . . . . . . . . . . . . . . . . . 22
3.2 The second-order Adams-Moulton-Bashforth method (AMB2) . . . . . . . . . . . . . 27
3.2.1 Efficiency of the scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.2 Unconditional stability of the AMB2 scheme . . . . . . . . . . . . . . . . . . 28
3.2.3 Uniform in time estimates for the AMB2 scheme . . . . . . . . . . . . . . . . 30
3.2.4 Uniform in time H1(Ω)-norm bound for the AMB2 scheme . . . . . . . . . . 32
3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 One type of third-order IMplicit-EXplicit (IMEX) method 42
4.1 Third-order Adams-Moulton-Bashforth (AMB3) scheme . . . . . . . . . . . . . . . . 42
4.2 Efficiency of the scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Important lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.4 Unconditional stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.5 Long-time stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.6 Error analysis of the semi-discrete scheme . . . . . . . . . . . . . . . . . . . . . . . . 52
4.7 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
iv
5 Uncertainty quantification of contaminant transport 61
5.1 Convection-Diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Numerical algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.3 Quantity of interest (QoI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3.1 Source of uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3.2 Quantity of interest for contaminant transport . . . . . . . . . . . . . . . . . 64
5.3.3 Karhunen-Loe`ve (KL) expansion . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.4 Methods for stochastic PDE with random input data . . . . . . . . . . . . . . . . . . 65
5.4.1 Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.4.2 Sparse grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.5.1 Numerical convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.5.2 Uncertainty quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6 Summary 74
Appendix
A Important inequalities 75
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
v
LIST OF TABLES
3.1 RelativeerrorandorderofaccuracywithrespecttothespatialgridsizehforExample
1 at t = 1 and with ∆t = h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 Same information as in Table 3.1 but for ∆t = h3.5/2. . . . . . . . . . . . . . . . . . . 36
3.3 Same information as in Table 3.1 but for ∆t = h2 . . . . . . . . . . . . . . . . . . . . 37
3.4 RelativeerrorandorderofaccuracywithrespecttothespatialgridsizehforExample
3 at t = 1 and with ∆t = h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1 RelativeerrorandorderofaccuracywithrespecttothespatialgridsizehforExample
1 at t = 1 and with ∆t = h. r = r defined by (4.47) . . . . . . . . . . . . 56
terminal 1/512,1
5.1 Numerical Error and Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 Comparison between Monte Carlo and Sparse grid Methods . . . . . . . . . . . . . . . 68
5.3 Comparison of QoI and QoI between different spatial grid sizes using Sparse Grid
1 2
Hermite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.4 Comparison of QoI and QoI between different spatial grid sizes using Sparse Grid
3 4
Hermite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.5 UncertaintyQuantificationofα independentlyusingSparseGridClenshaw-Curtis
BJSJ
for QoI and QoI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
1 2
5.6 UncertaintyQuantificationofinflowindependentlyusingSparseGridClenshaw-Curtis
for QoI and QoI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
1 2
5.7 Uncertainty Quantification of outflow independently using Sparse Grid Clenshaw-
Curtis for QoI and QoI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
1 2
5.8 UncertaintyQuantificationofα independentlyusingSparseGridClenshaw-Curtis
BJSJ
for QoI and QoI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3 4
5.9 UncertaintyQuantificationofinflowindependentlyusingSparseGridClenshaw-Curtis
for QoI and QoI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3 4
5.10 Uncertainty Quantification of outflow independently using Sparse Grid Clenshaw-
Curtis for QoI and QoI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3 4
vi
LIST OF FIGURES
2.1 The physical domain consisting of a porous media Ω and a free-flow conduit Ω . . . . 5
p f
3.1 Relative error for ϕ in Example 3 for BDF2 (left) and AMB2 (right) up to t = 100 for
h = 1/64. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Same information as for Figure 3.1 but for u . . . . . . . . . . . . . . . . . . . . . . . 38
f
3.3 Same information as for Figure 3.1 but for p. . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Energy E over time t. ν = S = g = α = 1, γ = γ = g/5, K = 10 6 and h = 1/20 39
BJSJ p f −
3.5 Energy E over time t. S = 10 6, ν = g = α = 1, γ = γ = g/5, K = 10 6 and
− BJSJ p f −
h = 1/20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6 Energy E over time t. ν = S = g = α = 1, γ = γ = g/10, K = 10 6 and
BJSJ p f −
h = 1/20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.7 Energy E over time t. ν = 10 1, S = g = α = 1, γ = γ = g/5, K = 10 6 and
− BJSJ p f −
h = 1/20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1 Relative error for hydraulic head ϕ (left) and velocity ⃗u (right) up to t = 100 for
h = 1/64. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Relative error for pressure p up to t = 100 for h = 1/64. . . . . . . . . . . . . . . . . . 57
4.3 Long-time behavior of energy functional E +E (left) and E (right) . . . . . . . . . 57
ϕ u p
4.4 Long-time behavior of energy functional E +E , ν = 0.0001 . . . . . . . . . . . . . . 58
ϕ u
4.5 Long-time behavior of energy functional E , ν = 0.0001 . . . . . . . . . . . . . . . . . 58
p
4.6 Long-time behavior of energy functional E +E , K = 0.01I . . . . . . . . . . . . . . 58
ϕ u
4.7 Long-time behavior of energy functional E , K = 0.01I . . . . . . . . . . . . . . . . . 59
p
4.8 Long-time behavior of energy functional E +E , K = 0.01I, γ = γ = g/2 . . . . . 59
ϕ u f p
4.9 Long-time behavior of energy functional E , K = 0.01I, γ = γ = g/2 . . . . . . . . . 59
p f p
vii
ABSTRACT
The dissertation includes two parts. The first part consists of designing and analyzing high order
long-timeaccuratenumericalmethodsforStokes-Darcysystem. Weproposesecondandthird-order
efficient and long-time accurate numerical methods, called IMplicit-EXplicit methods (IMEX) for
the coupled Stokes-Darcy system. Although the original continuum Stokes-Darcy PDE system is
fullycoupled,ouralgorithmiscapableofdecouplingthesystemintotwosub-systemssothatasingle
Stokes and a single Darcy system can be computed in a parallel fashion without iteration. All the
schemes we proposed are proven to be unconditionally stable and long-time stable. The bound on
theerrorisuniform-in-time,whichisamongthefirstofthiskindforsecondandthird-ordermethods
of Stokes-Darcy system. Error estimates for the second order Backward-Differentiation scheme are
proved. The second part concerns the Uncertainty of Quantification (UQ) of the contaminant
transport. We compute the convection-diffusion equation with Streamline Upwind Petrov-Galerkin
(SUPG) method. The quantity of interest is acquired using Monte Carlo and Sparse Grid methods
in order to study the sensitivity with respect to the random parameters.
viii
CHAPTER 1
INTRODUCTION
Karst is a type of landscape which contains special rocks, such as limestone and dolomite. Due
to the chemical reaction involving carbon dioxide and water, rock will gradually dissolve so that
tunnels and vugs are formed. As long as the water supply is sufficient, the tunnels and vugs will
develop into large holes and various shapes of rocks. The dissolution process will stop when there
is no more water, and the rock’s shape change will reach a steady state. A typical example is the
Yosemite area in California.
A karst aquifer is a type of aquifer with this unique karst landscape and is important to our
daily lives. According to a geological survey, about 40% of drinking water in the United States
comes from a karst aquifer. Moreover, the number is around 90% in Florida [35]. Karst aquifers
are very vulnerable to pollution due to human activities, including litters, pesticides and polluted
rain. Thus, it is crucial to study the flow motion and contaminant transport in a karst aquifer [37].
Due to the difficulty studying the karst aquifer, including heterogeneity of the porous media
part and the complicated conduit geometry, many hydrogeologists neither have the appropriate
mathematical models to compute nor enough information to study a karst aquifer. Thus, it is very
difficult to provide policy makers with useful suggestions and guidance about utilizing, developing
and protecting karst aquifers. Since contaminants are transported very slowly from one location to
another, we need to compute the numerical solution for large time.
However, the standard stability does not preclude exponential growth of the solution on a finite
interval. This may become a problem even with exact initial conditions due to rounding error. If
the terminal time is very large, there is no way to trust the numerical results as it will most likely
be ruined by amplified rounding errors. Besides accuracy, the computational cost is also a concern
for long-time simulations. In this dissertation, several high-order efficient and long-time accurate
schemes are proposed and analyzed. High-order schemes usually enable us to use a larger time step
than low-order schemes while maintaining the same accuracy, which also reduce the computational
costs.
1