PREFACE TO THE SECOND EDITION |
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xi | (4) |
PREFACE TO THE FIRST EDITION |
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xv | |
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1. SHORT REVIEW OF LINEAR ALGEBRA |
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1 | (13) |
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1.1. Introduction and Notation |
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1 | (2) |
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1.2. Gauss Elimination and LU Decomposition |
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3 | (4) |
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1.3. Tridiagonal and Other Banded Systems |
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7 | (2) |
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9 | (2) |
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11 | (3) |
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14 | (29) |
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2.1. Lagrange Interpolation |
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15 | (12) |
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15 | (2) |
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17 | (1) |
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2.1.3. Divided Differences |
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18 | (1) |
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19 | (5) |
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2.1.5. Piecewise Polynomial Interpolation |
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24 | (2) |
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26 | (1) |
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2.2. Hermite Interpolation |
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27 | (1) |
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28 | (9) |
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2.3.1. Definition and Development |
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28 | (4) |
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2.3.2. Examples and Programs |
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32 | (3) |
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35 | (1) |
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36 | (1) |
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37 | (1) |
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2.5. Parametric and Multidimensional Interpolation-Computer Graphics |
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38 | (3) |
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2.5.1. Parametric Interpolation |
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39 | (1) |
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2.5.2. Multidimensional Interpolation |
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40 | (1) |
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2.5.3. Graphics and Design |
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40 | (1) |
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41 | (2) |
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43 | (32) |
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3.1. Newton-Cotes Formulas |
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45 | (5) |
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3.2. Richardson Extrapolation and Error Estimation |
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50 | (3) |
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53 | (4) |
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57 | (8) |
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65 | (5) |
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70 | (2) |
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72 | (1) |
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3.7.1. Integration by Parts |
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72 | (1) |
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3.7.2. Singularity Subtraction |
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72 | (1) |
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73 | (1) |
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73 | (2) |
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4. ORDINARY DIFFERENTIAL EQUATIONS: I. INITIAL VALUE PROBLEMS |
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75 | (63) |
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4.1. Numerical Differentiation |
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77 | (9) |
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77 | (4) |
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81 | (4) |
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4.1.3. Numerical Integration |
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85 | (1) |
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86 | (1) |
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4.3. Euler Explicit Method |
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87 | (3) |
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90 | (10) |
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4.5. Backward or Implicit Euler Method |
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100 | (3) |
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4.6. Error Estimation and Accuracy Improvement |
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103 | (6) |
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104 | (1) |
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4.6.2. Richardson Extrapolation |
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105 | (1) |
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106 | (2) |
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108 | (1) |
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4.7. Predictor-Corrector Methods |
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109 | (2) |
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111 | (5) |
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116 | (7) |
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4.10. Choice of Method: Automatic Error Control |
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123 | (1) |
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4.11. Systems of Equations-Stiffness |
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124 | (9) |
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4.11.1. Treatment of Systems of Ordinary Differential Equations |
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124 | (1) |
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125 | (2) |
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4.11.3. Numerical Methods for Stiff Problems |
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127 | (4) |
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4.11.4. Splitting Methods |
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131 | (2) |
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4.11.5. Variable-Step-Size Methods |
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133 | (1) |
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4.12. Inherent Instability |
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133 | (1) |
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134 | (1) |
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134 | (4) |
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5. ORDINARY DIFFERENTIAL EQUATIONS: II. BOUNDARY VALUE PROBLEMS |
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138 | (44) |
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139 | (11) |
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5.2. Direct Methods: Introduction |
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150 | (5) |
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5.3. Higher-Order Direct Methods |
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155 | (4) |
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159 | (2) |
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161 | (5) |
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5.5.1. Finite Difference Approximations |
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162 | (1) |
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5.5.2. Coordinate Transformations |
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163 | (3) |
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5.6. Finite Element Methods |
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166 | (3) |
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169 | (3) |
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172 | (8) |
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174 | (4) |
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178 | (2) |
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180 | (2) |
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6. PARTIAL DIFFERENTIAL EQUATIONS: I. PARABOLIC EQUATIONS |
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182 | (46) |
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6.1. Classification of Partial Differential Equations |
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183 | (4) |
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183 | (4) |
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187 | (9) |
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6.3. Crank-Nicolson Method |
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196 | (6) |
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6.4. Dufort-Frankel Method |
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202 | (3) |
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205 | (2) |
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6.6. Second-Order Backward Method |
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207 | (1) |
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6.7. Higher-Order Methods |
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208 | (2) |
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6.8. Two and Three Spatial Dimensions: Alternating Direction Implicit Methods |
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210 | (10) |
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6.8.1. Heat Equation in Two Dimensions |
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210 | (3) |
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6.8.2. Peaceman-Rachford Method |
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213 | (2) |
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6.8.3. Approximate Factorization |
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215 | (2) |
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6.8.4. Other Splitting Methods |
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217 | (3) |
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6.9. Other Coordinate Systems |
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220 | (4) |
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224 | (2) |
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6.11. Final Remarks-Other Methods |
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226 | (1) |
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226 | (2) |
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7. PARTIAL DIFFERENTIAL EQUATIONS: II. ELLIPTIC EQUATIONS |
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228 | (88) |
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229 | (8) |
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7.1.1. Finite Differences |
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229 | (3) |
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7.1.2. Finite Volume Approximations |
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232 | (2) |
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7.1.3. Boundary Conditions |
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234 | (1) |
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7.1.4. System of Equations |
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235 | (2) |
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237 | (1) |
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7.2. Introduction to Iterative Methods and Their Properties |
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237 | (7) |
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7.2.1. Construction of Iterative Methods |
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237 | (2) |
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7.2.2. Errors in Iterative Methods |
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239 | (1) |
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240 | (1) |
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7.2.4. Stopping Criterion |
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241 | (2) |
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7.2.5. Estimation of Discretization Error |
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243 | (1) |
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244 | (6) |
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244 | (1) |
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244 | (1) |
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7.3.3. Connection to Heat Equation |
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245 | (1) |
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246 | (4) |
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250 | (3) |
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7.5. Line Relaxation Method |
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253 | (1) |
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7.6. Successive Overrelaxation |
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254 | (9) |
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255 | (2) |
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7.6.2. Point Successive Overrelaxation |
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257 | (4) |
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7.6.3. Successive Line Overrelaxation |
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261 | (2) |
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7.7. Alternating Direction Implicit Methods |
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263 | (5) |
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7.8. Incomplete LU Decomposition: Stone's Method |
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268 | (5) |
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7.9. Methods for Parallel Computers |
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273 | (3) |
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7.9.1. Red-Black Gauss-Seidel Method |
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274 | (2) |
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7.9.2. Parallelization of Other Methods |
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276 | (1) |
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276 | (7) |
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7.11. Conjugate Gradient Methods |
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283 | (7) |
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283 | (2) |
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285 | (1) |
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7.11.3. Biconjugate Gradients and CGSTAB |
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286 | (4) |
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290 | (2) |
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7.13. Finite Element Methods |
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292 | (5) |
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7.14. Discrete Fourier Transforms |
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297 | (9) |
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7.14.1. Review of Fourier Series |
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297 | (2) |
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7.14.2. Discrete Fourier Series |
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299 | (3) |
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7.14.3. Spectral Differentiation |
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302 | (1) |
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7.14.4. Fast Fourier Transform |
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303 | (3) |
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7.15. Fourier or Spectral Methods |
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306 | (2) |
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7.16. Boundary Integral Methods |
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308 | (3) |
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7.17. Finite Differences in Complex Geometry |
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311 | (2) |
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313 | (3) |
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8. PARTIAL DIFFERENTIAL EQUATIONS: III. HYPERBOLIC EQUATIONS |
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316 | (47) |
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317 | (8) |
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8.1.1. Quasi-Linear First-Order Equations |
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317 | (1) |
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8.1.2. Characteristics of Second-Order Equations |
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318 | (4) |
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8.1.3. Nonlinear Equations and Shocks |
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322 | (3) |
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8.2. Method of Characteristics |
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325 | (15) |
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8.2.1. First-Order Equations |
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325 | (2) |
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8.2.2. Second-Order Equations |
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327 | (10) |
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8.2.3. Method of Characteristics on Cartesian Grids |
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337 | (3) |
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340 | (8) |
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8.3.1. Explicit Central Difference Methods |
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341 | (2) |
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343 | (2) |
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8.3.3. Lax-Wendroff Method |
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345 | (3) |
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348 | (4) |
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352 | (9) |
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8.5.1. Explicit Split Methods |
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353 | (1) |
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8.5.2. Convection in Two Dimensions |
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354 | (6) |
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8.5.3. Implicit Split Methods |
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360 | (1) |
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361 | (2) |
APPENDIX A: LIST OF COMPUTER CODES |
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363 | (3) |
APPENDIX B: ANNOTATED BIBLIOGRAPHY |
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366 | (5) |
APPENDIX C: NOTE ON THE NEWTON-RAPHSON METHOD |
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371 | (2) |
INDEX |
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373 | |