Geiser J. Decomposition Methods for Differential Equations: Theory and Applications
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Decomposition
Methods for
Dierential
Equations
eory and Applications
© 2009 by Taylor & Francis Group, LLC
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© 2009 by Taylor & Francis Group, LLC
Decomposition
Methods for
Dierential
Equations
eory and Applications
Juergen Geiser
© 2009 by Taylor & Francis Group, LLC
CRC Press
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Preface
The monograph was written at the Department of Mathematics, Humboldt-
Universit¨at zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany.
In the monograph we discuss the decomposition methods with respect to
the discretisation method for evolution equations. The considered evolution
equations are systems of parabolic and hyperbolic equations. We deal with
convection-diffusion-reaction equations, heat-equations and wave-equations.
The applications are transport-reaction models, heat transfer models, micro-
magnetics and elastic-wave propagation models.
We propose efficient time and spatial decomposition methods, with respect
to the underlying discretization and solver methods. The main advantage
is efficiency in the computational- and memory-resources based on the sim-
plification in the decoupled equations. Further we propose the combination
of time and spatial decomposition and discretization methods, so that the
splitting and discretization error can be neglected with respect to the higher
order methods. One benefit of the iterative splitting method is to have all
operators in their iteration steps, and such behavior preserves the physics in
each equations.
We discuss in the theoretical part the stability and consistency of the de-
composition methods and in the practical part the application of the methods
with their benefits in decomposing test- and real-life-problems.
This work has been accompanied and supported by many coworkers and
colleagues and I would like to thank all of them.
First, I gratefully want to thank my colleagues and coworkers at the
Humboldt-University of Berlin for their fruitful discussions and ideas. In
particular, I wish to thank Prof.Q. Sheng, Prof.I. Farago, Dr.R. M¨uller and
the student workers L. Noack, V. Schlosshauer and R. Huth for their support
and ideas.
My special thanks to my wife Andrea and my daughter Lilli who have
always supported and encouraged me.
Berlin, December 2008 J¨urgen Geiser
vii
© 2009 by Taylor & Francis Group, LLC
Contents
List of Figures xvii
List of Tables xxi
Introduction 1
1 Modeling: Multi-Physics Problems 5
1.1 Introduction ........................... 5
1.2 Models for Multi-Physics Problems . . . . . . . . . . . . . . 5
1.3 Examples for Multi-Physics Problems . . . . . . . . . . . . . 6
1.3.1 WasteDisposal...................... 6
1.3.2 CrystalGrowth ..................... 8
1.3.3 Elastic Wave Propagation . . . . . . . . . . . . . . . 10
1.3.4 MagneticTrilayers .................... 11
2 Abstract Decomposition and Discretization Methods 13
2.1 Decomposition ......................... 13
2.1.1 Decomposition of the Evolution Equations . . . . . . . 14
2.1.2 PhysicalDecomposition ................. 14
2.1.2.1 Direct Decoupling Method . . . . . . . . . . 14
2.1.2.2 Decoupling Method with Respect to Numeri-
calMethods .................. 15
2.1.3 Mathematical Decomposition . . . . . . . . . . . . . . 16
2.1.3.1 Decomposition with Respect to the Maximal
Eigenvalues................... 16
2.1.3.2 Decomposition with Respect to Appropriate
Norms...................... 18
2.1.3.3 Abstract Decomposition Based on the Spec-
trum of the Operators (Symmetric Case) . . 18
2.1.4 Numerical Analysis of the Decomposition Methods . . 20
2.1.5 DecouplingIndex .................... 20
2.2 Discretization .......................... 21
2.2.1 GalerkinMethod .................... 21
2.2.2 Parabolic Differential Equation Applied with the Galerkin
Method .......................... 22
2.2.3 Hyperbolic Differential Equation Applied with the Galerkin
Method .......................... 24
ix
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x
2.2.4 OperatorEquation ................... 25
2.2.5 SemigroupTheory .................... 26
2.2.6 Classification of Semigroups . . . . . . . . . . . . . . . 26
2.2.7 Abstract Linear Parabolic Equations . . . . . . . . . . 28
2.2.8 Abstract Linear Hyperbolic Equations . . . . . . . . . 29
2.2.9 Nonlinear Equations . . . . . . . . . . . . . . . . . . . 30
3 Time-Decomposition Methods for Parabolic Equations for
Parabolic Equations 33
3.1 Introduction for the Splitting Methods . . . . . . . . . . . . 34
3.1.1 Classical Operator-Splitting Methods . . . . . . . . . . 35
3.1.2 Sequential Operator-Splitting Method . . . . . . . . . 35
3.1.3 Symmetrically Weighted Sequential Operator-Splitting 36
3.1.4 Strang-Marchuk Operator-Splitting Method . . . . . . 37
3.1.5 Higher-Order Splitting Method . . . . . . . . . . . . . 37
3.2 Iterative Operator-Splitting Methods for Bounded Operators 40
3.2.1 Physical Operator-Splitting Methods . . . . . . . . . 40
3.2.1.1 Example for Decoupling a Differential Equa-
tion ....................... 41
3.2.2 Introduction to the Iterative Operator-Splitting Meth-
ods............................. 42
3.2.3 Consistency Analysis of the Iterative Operator-Splitting
Method .......................... 44
3.2.3.1 Local Error Analysis of the Iterative Operator-
SplittingMethod ............... 44
3.2.3.2 Increase of the Order of Accuracy with Im-
provedInitialFunctions............ 48
3.2.4 Quasi-Linear Iterative Operator-Splitting Methods with
Bounded Operators . . . . . . . . . . . . . . . . . . . 51
3.2.5 Stability Analysis for the Linear and Quasi-Linear
Operator-SplittingMethod ............... 55
3.2.5.1 Stability for the Linear Weighted Iterative Split-
tingMethod .................. 55
3.2.5.2 A Posteriori Error Estimates for the Bounded
Operators and the Linear Operator-Splitting
Method .................... 60
3.2.5.3 Stability for the Quasi-Linear Iterative Split-
tingMethod .................. 61
3.2.6 Detection of the Correct Splitting Operators . . . . . 63
3.3 Iterative Operator-Splitting Methods for Unbounded Opera-
tors ................................ 64
3.3.1 Consistency Analysis for Unbounded Operators . . . . 66
3.3.1.1 Diffusion-Reaction Equation . . . . . . . . . 66
3.3.1.2 Convection-Diffusion-Reaction Equation . . . 68
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xi
4 Decomposition Methods for Hyperbolic Equations 73
4.1 Introduction for the Splitting Methods . . . . . . . . . . . . 73
4.2 ADIMethodsandLODMethods ............... 75
4.2.1 ClassicalADIMethod .................. 76
4.2.2 LODMethod ...................... 77
4.2.2.1 Stability and Consistency of the Fourth-Order
SplittingMethod................ 78
4.2.3 LOD Method Applied to a System of Wave Equations 80
4.2.3.1 Basic Numerical Methods . . . . . . . . . . . 80
4.2.3.2 Fourth-Order Splitting Method . . . . . . . . 82
4.2.3.3 Stability and Consistency of the Fourth-Order
Splitting Method for the System of Wave Equa-
tions....................... 83
4.3 Iterative Operator-Splitting Methods for Wave Equations . 87
4.3.1 Stability and Consistency Theory for the Iterative
Operator-SplittingMethod ............... 89
4.3.1.1 Consistency for the Iterative Operator-Splitting
MethodforWaveEquations ......... 89
4.3.1.2 Stability for the Iterative Operator-Splitting
MethodforWaveEquations ......... 91
4.4 Parallelization of Time Decomposition Methods . . . . . . . 94
4.4.1 Windowing ........................ 94
4.4.2 Block-WiseClustering.................. 94
4.5 Nonlinear Iterative Operator-Splitting Methods . . . . . . . 96
4.5.1 Linearization with the Waveform Relaxation . . . . . 96
5 Spatial Decomposition Methods 97
5.1 Domain Decomposition Methods Based on Iterative Operator-
SplittingMethods ........................ 97
5.1.1 The Nonoverlapping Spatiotemporal Iterative Splitting
Method .......................... 98
5.1.2 The Overlapping Spatiotemporal Iterative Splitting
Method .......................... 100
5.1.3 Error Analysis and Convergence of the Combined
Method .......................... 102
5.2 Schwarz Waveform-Relaxation Methods . . . . . . . . . . . . 106
5.2.1 Introduction ....................... 106
5.3 Overlapping Schwarz Waveform Relaxation for the Solution of
Convection-Diffusion-ReactionEquation............ 107
5.3.1 Overlapping Schwarz Waveform Relaxation for the Scalar
Convection-Diffusion-Reaction Equation . . . . . . . . 108
5.3.2 Overlapping Schwarz Waveform Relaxation for Decou-
pled System of Convection-Diffusion-Reaction Equation 112
5.3.3 Overlapping Schwarz Waveform Relaxation for Coupled
System of Convection-Diffusion-Reaction Equation . . 115
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