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References [1] Assyr Abdulle. Fourth order Chebyshev methods with recurrence relation. SIAM Journal on Scientific Computing, 23(6):2041–2054, 2002. [2] G. M. Amdahl. Validity of the single-processor approach to achieving large scale computing capabilities. In AFIPS Conference Proceedings vol. 30, Altantic City, NJ, April 18-20, 1967, pages 483–485. AFIPS, 1967. [3] L. Armijo. Minimization of functions having Lipschitz-continuous first partial derivatives. Pacific J. Math., 16(6):1–3, 1966. [4] Anton Arnold and Francis Nier. Numerical analysis of the deterministic particle method applied to the Wigner equation. Mathematics of Computation, 58:645– 669, 1992. [5] N. N. Bogolubov and N. N. Bogolubov Jr. Introduction to Quantum Statistical Mechanics. World Scientific Publishing, Singapore, 1982. [6] P. N. Brown, G. D. Byrne, and A. C. Hindmarsh. VODE, a variable-coefficient 117
REFERENCES 118 ODE solver. Technical Report UCRL-98412, Lawrence Livermore National Lab- oratory, June 1988. [7] P. N. Brown and A. C. Hindmarsh. Reduced storage matrix methods in stiff ODE problems. Technical Report UCRL-95088, Lawrence Livermore National Laboratory, June 1987. [8] F.A. Buot and K. I. Jensen. Lattice Weyl-Wigner formulation of exact many- body quantum-transport theory and applications to novel solid-state quantum- based devices. Phys. Rev. B, 42:9429–9438, 1990. [9] E. Burroughs, L. Romero, R. Lehoucq, and A. Salinger. Linear stability of flow in a differentially heated cavity via large-scale eigenvalue calculations. International Journal of Numerical Methods for Heat and Fluid Flow, 14(4):803–822, 2004. [10] S. L. Campbell, I. C. F. Ipsen, C. T. Kelley, and C. D. M eyer. GMRES and the minimum polynomial. BIT, 36(4):664–675, 1996. [11] C. Cercignani. Theory and Application of the Boltzmann Equation. Scottish Academic Press, 1975. [12] Norman G. Einspruch and William R. Frensley, editors. Heterostructures and Quantum Devices, volume24ofVLSI Electronics: Microstructure Science. Aca- demic Press, San Diego, CA, 1994.
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 and 8:
Table of Contents List of Tables ix
Page 9 and 10:
4.3 ApplicationofSchauder’sFixedP
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 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
Page 53 and 54:
Page 55 and 56:
Page 57 and 58:
Chapter 3 Bifurcation Analysis 3.1
Page 59 and 60:
CHAPTER 3. BIFURCATION ANALYSIS 46
Page 61 and 62:
Page 63 and 64:
Page 65 and 66:
Page 67 and 68:
Page 69 and 70:
Page 71 and 72:
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
Page 79 and 80: CHAPTER 3. BIFURCATION ANALYSIS 66
Page 87 and 88: Chapter 4 Theory 4.1 Steady-State T
Page 89 and 90: CHAPTER 4. THEORY 76 If we write ou
Page 91 and 92: CHAPTER 4. THEORY 78 So if we write
Page 93 and 94: CHAPTER 4. THEORY 80 Combining Equa
Page 95 and 96: CHAPTER 4. THEORY 82 We now estimat
Page 97 and 98: CHAPTER 4. THEORY 84 pendent of x,
Page 99 and 100: CHAPTER 4. THEORY 86 interval we ha
Page 101 and 102: CHAPTER 4. THEORY 88 So an estimate
Page 103 and 104: CHAPTER 4. THEORY 90 We want to cho
Page 105 and 106: CHAPTER 4. THEORY 92 The exact same
Page 107 and 108: CHAPTER 4. THEORY 94 Since un → u
Page 109 and 110: CHAPTER 4. THEORY 96 thereexistsaco
Page 111 and 112: CHAPTER 4. THEORY 98 For 0
Page 113 and 114: CHAPTER 4. THEORY 100 denote this d
Page 115 and 116: CHAPTER 4. THEORY 102 Now, we can t
Page 117 and 118: CHAPTER 4. THEORY 104 operator. 4.2
Page 119 and 120: CHAPTER 4. THEORY 106 we have at th
Page 121 and 122: CHAPTER 4. THEORY 108 we have at th
Page 123 and 124: CHAPTER 4. THEORY 110 4.3 Applicati
Page 125 and 126: CHAPTER 4. THEORY 112 We need to fi
Page 127 and 128: Chapter 5 Conclusion In this work,
Page 129: CHAPTER 5. CONCLUSION 116 • Findi
Page 133 and 134: REFERENCES 120 [21] C. T. Kelley an
Page 135 and 136: REFERENCES 122 [37] Homer F. Walker
Page 137 and 138: APPENDIX A. NOTATION 124 ˜B Genera
Page 139 and 140: APPENDIX A. NOTATION 126 k Wave num
Page 141 and 142: APPENDIX A. NOTATION 128 ∆t Incre
Page 143 and 144: APPENDIX A. NOTATION 130 ɛc ɛk ɛ
Page 145 and 146: APPENDIX B. TRILINOS 132 user-given
Page 147 and 148: APPENDIX B. TRILINOS 134 Trilinos,
Page 149 and 150: APPENDIX B. TRILINOS 136 Minimum Re
Page 151 and 152: APPENDIX B. TRILINOS 138 A very use
Page 153 and 154: Appendix C Guide to RTD Simulation
Page 155 and 156: APPENDIX C. GUIDE TO RTD SIMULATION
Page 163 and 164: Appendix D GMRES and Arnoldi Iterat
Page 165 and 166: APPENDIX D. GMRES AND ARNOLDI ITERA
Page 167 and 168: APPENDIX D. GMRES AND ARNOLDI ITERA
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