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2.4 Preconditioning.............................. 37 2.5 GlobalConvergence............................ 38 2.6 PreconditionerDevelopment....................... 39 2.7 Calculating Equilibrium Wigner Function . . . . . . . . . . . . . . . . 40 3 Bifurcation Analysis 44 3.1 Stability of Nonlinear Dynamics . . . . . . . . . . . . . . . . . . . . . 44 3.2 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3 ContinuationMethods .......................... 50 3.4 Linear Solver Methods for Pseudo Arc-Length Continuation . . . . . 53 3.5 BorderingAlgorithms........................... 55 3.6 HouseholderContinuation ........................ 56 3.7 ParallelizationofSimulator ....................... 57 3.8 Linear Solver Comparison for Pseudo Arc-Length Continuation . . . . 62 3.9 Results................................... 64 3.10CorrelationLengthStudy ........................ 68 3.10.1 CurrentVoltageRelationship .................. 68 3.10.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 72 4 Theory 74 4.1 Steady-StateTheory ........................... 74 4.2 Infinite-DimesionalAnalysisofGMRES................. 104 vi
4.3 ApplicationofSchauder’sFixedPointTheorem ............ 110 5 Conclusion 114 References 117 A Notation 123 B Trilinos 131 B.1 Introduction................................ 131 B.2 Procedural and Object-Oriented Programming . . . . . . . . . . . . . 132 B.3 Trilinos Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 B.3.1 Epetra............................... 134 B.3.2 LOCA............................... 135 B.3.3 NOX................................ 135 B.3.4 AztecOO.............................. 135 B.3.5 Anasazi .............................. 136 B.4 Contribution................................ 136 C Guide to RTD Simulation Code 140 C.1 InitializationCode ............................ 141 C.1.1 equil.f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 C.1.2 transi.f............................... 141 C.1.3 wprockinit.f ............................ 141 vii
Page 1 and 2: Numerical Methods for the Wigner-Po
Page 3 and 4: NUMERICAL METHODS FOR THE WIGNER-PO
Page 5 and 6: Acknowledgments First and foremost,
Page 7: Table of Contents List of Tables ix
Page 11 and 12: List of Tables 1.1 PhysicalConstant
Page 13 and 14: 3.11 I-V Curve with 30 Percent Redu
Page 15 and 16: CHAPTER 1. OVERVIEW 2 being heavily
Page 17 and 18: CHAPTER 1. OVERVIEW 4 Doping Spacer
Page 19 and 20: CHAPTER 1. OVERVIEW 6 and identfyin
Page 21 and 22: CHAPTER 1. OVERVIEW 8 for predictin
Page 23 and 24: CHAPTER 1. OVERVIEW 10 by Schrödin
Page 25 and 26: CHAPTER 1. OVERVIEW 12 ranging term
Page 27 and 28: CHAPTER 1. OVERVIEW 14 But because
Page 29 and 30: CHAPTER 1. OVERVIEW 16 = h 2π ,we
Page 31 and 32: CHAPTER 1. OVERVIEW 18 Table 1.1: P
Page 33 and 34: CHAPTER 1. OVERVIEW 20 where q is t
Page 35 and 36: CHAPTER 1. OVERVIEW 22 get K(f)
Page 37 and 38: CHAPTER 1. OVERVIEW 24 rule. The pr
Page 39 and 40: CHAPTER 1. OVERVIEW 26 The weights
Page 41 and 42: Chapter 2 Temporal Integration 2.1
Page 43 and 44: CHAPTER 2. TEMPORAL INTEGRATION 30
Page 57 and 58: Chapter 3 Bifurcation Analysis 3.1
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CHAPTER 3. BIFURCATION ANALYSIS 46
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Chapter 4 Theory 4.1 Steady-State T
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CHAPTER 4. THEORY 76 If we write ou
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CHAPTER 4. THEORY 78 So if we write
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CHAPTER 4. THEORY 80 Combining Equa
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CHAPTER 4. THEORY 82 We now estimat
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CHAPTER 4. THEORY 84 pendent of x,
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CHAPTER 4. THEORY 86 interval we ha
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CHAPTER 4. THEORY 88 So an estimate
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CHAPTER 4. THEORY 90 We want to cho
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CHAPTER 4. THEORY 92 The exact same
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CHAPTER 4. THEORY 94 Since un → u
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CHAPTER 4. THEORY 96 thereexistsaco
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CHAPTER 4. THEORY 98 For 0
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CHAPTER 4. THEORY 100 denote this d
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CHAPTER 4. THEORY 102 Now, we can t
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CHAPTER 4. THEORY 104 operator. 4.2
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CHAPTER 4. THEORY 106 we have at th
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CHAPTER 4. THEORY 108 we have at th
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CHAPTER 4. THEORY 110 4.3 Applicati
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CHAPTER 4. THEORY 112 We need to fi
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Chapter 5 Conclusion In this work,
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CHAPTER 5. CONCLUSION 116 • Findi
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REFERENCES 118 ODE solver. Technica
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REFERENCES 120 [21] C. T. Kelley an
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REFERENCES 122 [37] Homer F. Walker
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APPENDIX A. NOTATION 124 ˜B Genera
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APPENDIX A. NOTATION 126 k Wave num
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APPENDIX A. NOTATION 128 ∆t Incre
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APPENDIX A. NOTATION 130 ɛc ɛk ɛ
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APPENDIX B. TRILINOS 132 user-given
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APPENDIX B. TRILINOS 134 Trilinos,
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APPENDIX B. TRILINOS 136 Minimum Re
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APPENDIX B. TRILINOS 138 A very use
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Appendix C Guide to RTD Simulation
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APPENDIX C. GUIDE TO RTD SIMULATION
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Appendix D GMRES and Arnoldi Iterat
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APPENDIX D. GMRES AND ARNOLDI ITERA
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APPENDIX D. GMRES AND ARNOLDI ITERA
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