Abstract

More documents

Recommendations

Info

CHAPTER 2. TEMPORAL INTEGRATION 31 The computations were performed on a Linux cluster of dual Xeon nodes with 2.8 GHz Intel Xeon processors maintained by the High Peformance and Grid Computing group at North Carolina State University. Table 2.1: Sweep In Bias Timing Nx Nk ROCK4 VODEPK 86 72 2hr10min 2hr17min 128 128 13 hr 48 min 13 hr 45 min 172 144 32 hr 23 min 29 hr 53 min We see that for the two coarsest grids, the integrators take about the same amount of time to finish the sweep in bias. At the finest grid, we see that VODEPK is slightly outperforming ROCK4. The grid Nx = 344, Nk = 288 was also tried, but after 3 days of computation, ROCK4 had only started on the eighth bias point while VODEPK had started on the ninth bias point. While this reinforces the idea that VODEPK can handle the finer grids better than ROCK4, it also emphasizes that using time-integration methods to understand the physics predicted by the Wigner- Poisson equations is too expensive. This motivates the alternative method of directly calculating steady-state solutions to the Wigner-Poisson equations, which is explained in Chapter 3.
CHAPTER 2. TEMPORAL INTEGRATION 32 2.2 Newton-Krylov Methods To solve the nonlinear equations that come from the implicit integrator and in finding the equilibrium Wigner distribution, f0, on different grids, we use Newton’s method. Given a nonlinear equation F : R M → R M and an initial iterate z0, Newton’s method is an iterative method that finds a root of F , that is find z ∗ ∈ R M such that F (z ∗ )=0, by the iteration: zm+1 = zm − F ′ (zm) −1 F (zm) =zm + sm where F ′ (zm) is the Jacobian of F at zm. The Newton step, sm, is then the solution of the linear equation: F ′ (zm)sm = −F (zm) Before we cite the convergence theorems of Newton’s method, we first want to define two types of convergence, q-linear convergence and q-quadractic convergence. A sequence {zm} ∈R M converges to z ∗ q-linearly if the sequence converges to z ∗ , and there exists a constant C>0 such that zm+1 − z ∗ ≤Czm − z ∗ . A sequence {zm} ∈R M converges to z ∗ q-quadratically if the sequence converges to z ∗ ,andthere exists a constant C>0 such that zm+1 − z ∗ ≤Czm − z ∗ 2 . The standard assumptions [19] for this problem are: • There exists a solution z ∗ ∈ R M . • The Jacobian F ′ (z) is Lipschitz continuous
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: CHAPTER 2. TEMPORAL INTEGRATION 30
Page 47 and 48: CHAPTER 2. TEMPORAL INTEGRATION 34
Page 57 and 58: Chapter 3 Bifurcation Analysis 3.1
Page 59 and 60: CHAPTER 3. BIFURCATION ANALYSIS 46
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 and 130:
CHAPTER 5. CONCLUSION 116 • Findi
Page 131 and 132:
REFERENCES 118 ODE solver. Technica
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 157 and 158:
Page 159 and 160:
Page 161 and 162:
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
show all

Abstract

Create successful ePaper yourself

Delete template?

Save as template?