Stability of BDF-ADI Discretizations

85 Pages·2017·3.16 MB·English

Stability of BDF-ADI Discretizations
Thesis by
Joa˜o Miguel Fel´ıcio dos Reis
In Partial Fulfillment of the Requirements
For the Degree of
Masters of Science
King Abdullah University of Science and Technology
Thuwal, Kingdom of Saudi Arabia
August, 2017
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EXAMINATION COMMITTEE PAGE
The thesis of Joa˜o Miguel Fel´ıcio dos Reis is approved by the examination committee
Committee Chairperson: Professor David Ketcheson
Committee Co-Chair: Professor Lajos L´oczi
Committee Members: Professor David Keyes, Professor Dominik Michels
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©August, 2017
Joa˜o Miguel Fel´ıcio dos Reis
All Rights Reserved
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ABSTRACT
We present new results on absolute stability for BDF-ADI (Backward differenti-
ation formula Alternating Direction Implicit) methods applied to a linear advection
and diffusion equations. Unconditional absolute stability of the BDF2-ADI method
is proven for advection and diffusion separately, as well as to the BDF3-ADI method
for the purely-diffusive case. Conditional absolute stability of the BDF4-ADI is also
proven for the purely-diffusive case, and stability regions for BDF3-ADI and BDF4-
ADI are given in terms of the PDE coefficients and numerical parameters. Lastly,
numerical experiments are presented to support the theoretical results and conjec-
tures. These experiments also suggest future work.
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ACKNOWLEDGEMENTS
Firstly, I would like to express my sincere gratitude to my advisors Prof. David
Ketcheson and Prof. Lajos L´oczi for the continuous support on this project, for
their patience, motivation, and immense knowledge. Their guidance helped me in
all the time of research and writing of this thesis. Besides my advisers, I would
like to thank the rest of my thesis committee: Prof. David Keyes and Prof. Dominik
Michels for their insightful comments. I would not have known KAUST and the many
opportunities that this university has if not for Prof. Diogo Gomes. I would like to
express my sincere appreciation for showing me a new window of possibilities. I thank
my dear colleague Yiannis Hadjimichael for the stimulating discussions, and also for
answering without hesitation my numerous questions for the last 20 months. Also, I
thank my friends within the many departments of KAUST for the precious support
over the many days of helpless research. In particular, I am grateful to Francisco
Berkemeier and Mariana Prazeres for all the fun and loyalty. Last but not the least,
I would like to thank my family: my parents and sister for supporting me spiritually
throughout writing this thesis and my life in general.
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TABLE OF CONTENTS
Examination Committee Page 2
Copyright 3
Abstract 4
Acknowledgements 5
List of Figures 8
List of Tables 10
1 Introduction 11
2 Stability of Numerical Methods 17
2.1 Schur-Cohn polynomial reduction . . . . . . . . . . . . . . . . . . . . 20
3 BDF-ADI Spectral Collocation Method 24
3.1 Spectral Collocation Methods . . . . . . . . . . . . . . . . . . . . . . 26
3.1.1 Fourier analysis in one dimension . . . . . . . . . . . . . . . . 27
3.1.2 Fourier analysis in two dimensions . . . . . . . . . . . . . . . . 30
3.1.3 A spatial discretization . . . . . . . . . . . . . . . . . . . . . . 31
3.1.4 Spectrum of a spectral differentiation matrix based on Fourier
collocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 BDF Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.1 Backward Differentiation Formula . . . . . . . . . . . . . . . . 35
3.2.2 Stability Region . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 BDF-ADI Spectral Collocation Methods . . . . . . . . . . . . . . . . 40
4 Main Results 43
4.1 Stability for BDF2-ADI . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.1.1 Absolute stability of BDF2-ADI for the parabolic case with
mixed term . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
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4.1.2 Absolute stability of BDF2-ADI for the hyperbolic case . . . . 47
4.1.3 Absolute stability of BDF2-ADI for the mixed case . . . . . . 48
4.2 Stability of BDF3-ADI . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3 Stability of BDF4-ADI . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.4 Stability of BDF3-ADI and BDF4-ADI for the parabolic equation case
with mixed term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 Numerical Implementations 60
5.1 Unconditional stability . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Conditional Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2.1 BDF4-ADI for the diffusion equation . . . . . . . . . . . . . . 63
5.2.2 Parabolic case with mixed term . . . . . . . . . . . . . . . . . 65
6 Concluding Remarks 68
References 71
Appendices 73
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LIST OF FIGURES
3.1 Regions of stability (in the complex-valued plane) of BDF methods for
different orders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 The real-valued stability regions of BDF methods for the hyperbolic
equation. This is a two-dimensional representation of the imaginary
axis of regions in Figure 3.1. The BDF1 and BDF2 methods are un-
conditionally stable. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1 Stability region for (A ,B ) of BDF3-ADI for the hyperbolic problem,
1 1
together with the lattice of scaled eigenvalues. . . . . . . . . . . . . . 50
4.2 IllustrationofconditionalstabilityofBDF4-ADIforthepureparabolic
problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Conditional stability of BDF4-ADI for the pure parabolic problem . . 52
4.4 Stability region for (A ,B ,F) of BDF3-ADI for the problem (4.2).
2 2
This region is symmetric with respect to the plane A = B . However,
2 2
it is not symmetric with respect to F = 0. . . . . . . . . . . . . . . . 54
4.5 Lattice for the parabolic problem. This lattice corresponds to the
scaled eigenvalues A ,B and F with α = β = γ = 1 and ∆t = 0.001.
2 2 2 2
Each direction has 64 collocation points. . . . . . . . . . . . . . . . . 55
4.6 This is a good indication that this choice of parameters yields a stable
solution for ∆t ≤ 0.001. . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.7 Slices of the stability region of BDF3-ADI for the parabolic problem
(4.13). This scheme yields an unstable solution for ∆t = 0.002. . . . 56
4.8 Stability region for (A ,B ,F) of BDF4-ADI for problem (4.2). . . . 57
2 2
4.9 StablesolutionoftheBDF4-ADIappliedtotheparabolicproblem(4.14). 58
4.10 The solution of BDF4-ADI for (4.14) becomes unstable if instead we
consider ∆t = 0.0012 . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.1 Convergence test of BDF2-ADI applied to problem (5.1). . . . . . . . 61
5.2 Numerical solution of BDF3-ADI for problem (5.2). . . . . . . . . . 62
5.3 Convergence test of BDF3-ADI for the problem (5.2). The final time
is T = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
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5.4 Stable numerical solution of BDF4-ADI for problem (5.2). . . . . . . 63
5.5 Convergence test of BDF4-ADI for the problem (5.2). The final time
is T = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.6 Unstable numerical solution of BDF4-ADI for problem (5.2). The nu-
merical solution at T = 1.5 corresponding to ∆t = 0.001 has an error
of 10−13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.7 Stable numerical solution of BDF4-ADI for the parabolic problem (4.12). 65
5.8 Convergence test of BDF3-ADI for the parabolic problem (4.12). The
final time is T = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.9 Unstable numerical solution of BDF3-ADI for the parabolic problem
(4.13). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.10 Numerical solutions for problem (4.14). A slight change of the time
step can change the behaviour of the numerical solution drastically.
The numerical solution is stable for ∆t = 0.001 and unstable for ∆t =
0.0012. The numerical solution at T = 3 corresponding to ∆t = 0.001
has an error of 10−16. . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.11 Convergence of BDF4-ADI for the parabolic equation with the mixed
term. This example refers to (4.14). For ∆t < 10−4 the floating point
errors originate instabilities. . . . . . . . . . . . . . . . . . . . . . . . 67
6.1 We can conjecture conditional stability of BDF3-ADI and BDF4-ADI
for the mixed problem. . . . . . . . . . . . . . . . . . . . . . . . . . 69
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LIST OF TABLES
3.1 Coefficients for BDF methods of order s, with s = 1,...,6 . . . . . . . 35