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1 | (33) |
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1 | (3) |
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First-Order Partial Differential Equations |
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4 | (8) |
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First-Order Quasilinear Partial Differential Equations |
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4 | (2) |
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Initial Value or Cauchy Problem |
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6 | (1) |
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Application of Characteristic Curves |
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7 | (4) |
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Nonlinear First-Order Partial Differential Equations |
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11 | (1) |
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Second-Order Partial Differential Equations |
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12 | (9) |
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Linear Second-Order Partial Differential Equations |
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12 | (5) |
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Classification and Canonical Form of Selected Partial Differential Equations |
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17 | (1) |
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Quasilinear Partial Differential Equations and Other Ideas |
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17 | (4) |
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Systems of First-Order PDEs |
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21 | (7) |
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First-Order and Second-Order PDEs |
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21 | (3) |
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24 | (2) |
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Applications of Characteristic Curves |
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26 | (2) |
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Initial and Boundary Conditions |
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28 | (6) |
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33 | (1) |
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Basic Concepts in the Finite Difference and Finite Element Methods |
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34 | (75) |
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34 | (1) |
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Finite Difference Approximations |
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34 | (15) |
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35 | (1) |
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36 | (2) |
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Operator Notation for u(x) |
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38 | (3) |
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Finite Difference Approximations in Two Dimensions |
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41 | (2) |
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43 | (6) |
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Introduction to Finite Element Approximations |
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49 | (55) |
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Method of Weighted Residuals |
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49 | (4) |
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Application of the Method of Weighted Residuals |
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53 | (7) |
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The Choice of Basis Functions |
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60 | (19) |
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Two-Dimensional Basis Functions |
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79 | (11) |
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90 | (14) |
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Relationship between Finite Element and Finite Difference Methods |
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104 | (5) |
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107 | (2) |
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Finite Elements on Irregular Subspaces |
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109 | (40) |
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109 | (1) |
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109 | (11) |
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The Linear Triangular Element |
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109 | (1) |
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110 | (1) |
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The Quadratic Triangular Element |
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110 | (6) |
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The Cubic Triangular Element |
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116 | (4) |
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Higher-Order Triangular Elements |
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120 | (1) |
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Isoparametric Finite Elements |
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120 | (17) |
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120 | (6) |
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126 | (3) |
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Isoparametric Serendipity Hermitian Elements |
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129 | (2) |
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Isoparametric Hermitian Elements in Normal and Tangential Coordinates |
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131 | (6) |
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137 | (4) |
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Three-Dimensional Elements |
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141 | (8) |
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148 | (1) |
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Parabolic Partial Differential Equations |
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149 | (206) |
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149 | (1) |
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Partial Differential Equations |
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149 | (2) |
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Well-Posed Partial Differential Equations |
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151 | (1) |
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Model Difference Approximations |
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151 | (2) |
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Well-Posed Difference Forms |
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153 | (1) |
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Derivation of Finite Difference Approximations |
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153 | (9) |
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The Classic Explicit Approximation |
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155 | (2) |
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The Dufort-Frankel Explicit Approximation |
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157 | (1) |
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The Richardson Explicit Approximation |
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158 | (1) |
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The Backwards Implicit Approximation |
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159 | (1) |
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The Crank-Nicolson Implicit Approximation |
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160 | (1) |
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The Variable-Weighted Implicit Approximation |
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161 | (1) |
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Consistency and Convergence |
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162 | (4) |
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166 | (20) |
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168 | (2) |
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170 | (9) |
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179 | (7) |
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186 | (27) |
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Influence of Lower-Order Terms |
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186 | (1) |
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187 | (3) |
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Predictor-Corrector Methods |
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190 | (2) |
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Asymmetric Approximations |
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192 | (7) |
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199 | (4) |
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203 | (8) |
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211 | (2) |
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Solution of Finite Difference Approximations |
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213 | (6) |
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Solution of Implicit Approximations |
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214 | (4) |
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Explicit versus Implicit Approximations |
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218 | (1) |
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219 | (15) |
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220 | (4) |
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224 | (2) |
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226 | (4) |
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230 | (1) |
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Composites of Different Approximations |
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231 | (3) |
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Finite Difference Approximations in Two Space Dimensions |
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234 | (31) |
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234 | (6) |
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240 | (1) |
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241 | (3) |
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Alternating Direction Explicit (ADE) Methods |
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244 | (1) |
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Alternating Direction Implicit (ADI) Methods |
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245 | (10) |
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LOD and Fractional Splitting Methods |
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255 | (6) |
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261 | (3) |
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264 | (1) |
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Three-Dimensional Problems |
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265 | (11) |
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266 | (6) |
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LOD and Fractional Splitting Methods |
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272 | (2) |
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274 | (2) |
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Finite Element Solution of Parabolic Partial Differential Equations |
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276 | (9) |
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Galerkin Approximation to the Model Parabolic Partial Differential Equation |
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277 | (3) |
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Approximation of the Time Derivative |
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280 | (2) |
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Approximation of the Time Derivative for Weakly Nonlinear Equations |
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282 | (3) |
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Finite Element Approximations in One Space Dimensions |
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285 | (24) |
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Formulation of the Galerkin Approximating Equations |
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285 | (4) |
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Linear Basis Function Approximation |
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289 | (5) |
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Higher-Degree Polynomial Basis Function Approximation |
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294 | (3) |
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Formulation Using the Dirac Delta Function |
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297 | (2) |
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Orthogonal Collocation Formulation |
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299 | (7) |
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Asymmetric Weighting Functions |
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306 | (3) |
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Finite Element Approximations in Two Space Dimensions |
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309 | (39) |
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Galerkin Approximation in Space and Time |
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309 | (5) |
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Galerkin Approximation in Space Finite Difference in Time |
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314 | (2) |
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Asymmetric Weighting Functions in Two Space Dimensions |
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316 | (5) |
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Lumped and Consistent Time Matrices |
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321 | (9) |
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Collocation Finite Element Formulation |
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330 | (9) |
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Treatment of Sources and Sinks |
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339 | (3) |
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Alternating Direction Formulation |
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342 | (6) |
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Finite Element Approximations in Three Space Dimensions |
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348 | (2) |
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348 | (2) |
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350 | (5) |
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351 | (4) |
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Elliptic Partial Differential Equations |
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355 | (131) |
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355 | (1) |
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355 | (5) |
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355 | (1) |
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356 | (2) |
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358 | (2) |
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Finite Difference Solutions in Two Space Dimensions |
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360 | (74) |
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Five-Point Approximations and Truncation Error |
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360 | (11) |
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Nine-Point Approximations and Truncation Error |
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371 | (2) |
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Approximations to the Biharmonic Equation |
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373 | (2) |
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Boundary Condition Approximations |
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375 | (2) |
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Matrix Form of Finite Difference Equations |
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377 | (6) |
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Direct Methods of Solution |
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383 | (2) |
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385 | (7) |
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Formulation of Point Iterative Methods |
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392 | (13) |
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Convergence of Point Iterative Methods |
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405 | (13) |
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Line and Block Iteration Methods |
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418 | (3) |
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421 | (9) |
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Acceleration and Semi-Iterative Overlays |
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430 | (4) |
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Finite Difference Solutions in Three Space Dimensions |
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434 | (7) |
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Finite Difference Approximations |
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435 | (2) |
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437 | (1) |
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437 | (4) |
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Finite Element Methods for Two Space Dimensions |
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441 | (20) |
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442 | (3) |
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445 | (4) |
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Collocation Approximation |
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449 | (4) |
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Mixed Finite Element Approximation |
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453 | (2) |
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Approximation of the Biharmonic Equation |
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455 | (6) |
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Boundary Integral Equation Methods |
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461 | (20) |
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461 | (4) |
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Boundary Element Formulation |
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465 | (2) |
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467 | (4) |
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Linear Interpolation Functions |
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471 | (2) |
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473 | (2) |
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475 | (3) |
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Combination of Finite Element and Boundary Integral Equation Methods |
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478 | (3) |
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Three-Dimensional Finite Element Simulation |
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481 | (1) |
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482 | (4) |
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482 | (4) |
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Hyperbolic Partial Differential Equations |
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486 | (185) |
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486 | (1) |
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Equations of Hyperbolic Type |
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487 | (2) |
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Finite Difference Solution of First-Order Scalar Hyperbolic Partial Differential Equations |
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489 | (37) |
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Stability, Truncation Error, and Other Features |
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490 | (7) |
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497 | (8) |
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Dissipation and Dispersion |
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505 | (19) |
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Hopscotch Methods and Mesh Refinement |
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524 | (2) |
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Finite Difference Solution of First-Order Vector Hyperbolic Partial Differential Equations |
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526 | (2) |
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Finite Difference Solution of First-Order Vector Conservative Hyperbolic Partial Differential Equations |
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528 | (11) |
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Finite Difference Solutions to Two-and Three-Dimensional Hyperbolic Partial Differential Equations |
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539 | (23) |
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Finite Difference Schemes |
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540 | (5) |
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Two-Step, ADI, and Strang-Type Algorithms |
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545 | (9) |
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Conservative Hyperbolic Partial Differential Equations |
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554 | (8) |
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Finite Difference Solution of Second-Order Model Hyperbolic Partial Differential Equations |
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562 | (27) |
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One-Space-Dimension Hyperbolic Partial Differential Equation |
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562 | (2) |
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564 | (5) |
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569 | (2) |
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Simultaneous First-Order Partial Differential Equations |
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571 | (6) |
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577 | (3) |
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Two-and Three-Space-Dimensional Hyperbolic Partial Differential Equations |
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580 | (3) |
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Implicit ADI and LOD Methods |
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583 | (6) |
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Finite Element Solution of First-Order Model Hyperbolic Partial Differential Equations |
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589 | (31) |
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589 | (5) |
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Asymmetric Weighting Function Approximation |
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594 | (4) |
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An H++ Galerkin Approximation |
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598 | (1) |
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Orthogonal Collocation Formulation |
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599 | (5) |
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Orthogonal Collocation with Asymmetric Bases |
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604 | (1) |
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Dissipation and Dispersion |
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605 | (15) |
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Finite Element Solution of Two-and Three-Space-Dimensional First-Order Hyperbolic Partial Differential Equations |
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620 | (5) |
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Galerkin Finite Element Formulation |
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620 | (2) |
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Orthogonal Collocation Formulation |
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622 | (3) |
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Finite Element Solution of First-Order Vector Hyperbolic Partial Differential Equations |
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625 | (20) |
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Galerkin Finite Element Formulation |
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626 | (1) |
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Dissipation and Dispersion |
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627 | (18) |
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Finite Element Solution of Two-and Three-Space-Dimensional First-Order Vector Hyperbolic Partial Differential Equations |
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645 | (10) |
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Galerkin Finite Element Formulation |
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645 | (3) |
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648 | (7) |
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Finite Element Solution of One-Space-Dimensional Second-Order Hyperbolic Partial Differential Equations |
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655 | (10) |
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Galerkin Finite Element Formulation |
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655 | (2) |
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657 | (6) |
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Dissipation and Dispersion |
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663 | (2) |
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Finite Element Solution of Two-and Three-Space-Dimensional Second-Order Hyperbolic Partial Differential Equations |
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665 | (2) |
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Galerkin Finite Element Formulation |
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665 | (2) |
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667 | (4) |
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667 | (4) |
Index |
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671 | |