Numerical solution of partial differential equations in science and engineering / Leon Lapidus, George F. Pinder.

Format:

Book

Author/Creator:

Lapidus, Leon.

Contributor:

Pinder, George Francis, 1942-

Language:

English

Subjects (All):

Science--Mathematics.

Science.

Engineering mathematics.

Differential equations, Partial--Numerical solutions.

Differential equations, Partial.

Physical Description:

677 pages : illustrations ; 24 cm

Place of Publication:

New York : Wiley, [1982]

Contents:

Chapter 1. Fundamental Concepts 1

1.1. First-Order Partial Differential Equations 4

1.1.1. First-Order Quasilinear Partial Differential Equations 4

1.1.2. Initial Value or Cauchy Problem 6

1.1.3. Application of Characteristic Curves 7

1.1.4. Nonlinear First-Order Partial Differential Equations 11

1.2. Second-Order Partial Differential Equations 12

1.2.1. Linear Second-Order Partial Differential Equations 12

1.2.2. Classification and Canonical Form of Selected Partial Differential Equations 17

1.2.3. Quasilinear Partial Differential Equations and Other Ideas 17

1.3. Systems of First-Order PDEs 21

1.3.1. First-Order and Second-Order PDEs 21

1.3.2. Characteristic Curves 24

1.3.3. Applications of Characteristic Curves 26

1.4. Initial and Boundary Conditions 28

Chapter 2. Basic Concepts in the Finite Difference and Finite Element Methods 34

2.1. Finite Difference Approximations 34

2.1.2. Taylor Series Expansions 36

2.1.3. Operator Notation for u(x) 38

2.1.4. Finite Difference Approximations in Two Dimensions 41

2.1.5. Additional Concepts 43

2.2. Introduction to Finite Element Approximations 49

2.2.1. Method of Weighted Residuals 49

2.2.2. Application of the Method of Weighted Residuals 53

2.2.3. The Choice of Basis Functions 60

2.2.4. Two-Dimensional Basis Functions 79

2.2.5. Approximating Equations 90

2.3. Relationship between Finite Element and Finite Difference Methods 104

Chapter 3. Finite Elements on Irregular Subspaces 109

3.1. Triangular Elements 109

3.1.1. The Linear Triangular Element 109

3.1.2. Area Coordinates 110

3.1.3. The Quadratic Triangular Element 110

3.1.4. The Cubic Triangular Element 116

3.1.5. Higher-Order Triangular Elements 120

3.2. Isoparametric Finite Elements 120

3.2.1. Transformation Functions 120

3.2.2. Numerical Integration 126

3.2.3. Isoparametric Serendipity Hermitian Elements 129

3.2.4. Isoparametric Hermitian Elements in Normal and Tangential Coordinates 131

3.3. Boundary Conditions 137

3.4. Three-Dimensional Elements 141

Chapter 4. Parabolic Partial Differential Equations 149

4.1. Partial Differential Equations 149

4.1.1. Well-Posed Partial Differential Equations 151

4.2. Model Difference Approximations 151

4.2.1. Well-Posed Difference Forms 153

4.3. Derivation of Finite Difference Approximations 153

4.3.1. The Classic Explicit Approximation 155

4.3.2. The Dufort-Frankel Explicit Approximation 157

4.3.3. The Richardson Explicit Approximation 158

4.3.4. The Backwards Implicit Approximation 159

4.3.5. The Crank-Nicolson Implicit Approximation 160

4.3.6. The Variable-Weighted Implicit Approximation 161

4.4. Consistency and Convergence 162

4.5. Stability 166

4.5.1. Heuristic Stability 168

4.5.2. Von Neumann Stability 170

4.5.3. Matrix Stability 179

4.6. Some Extensions 186

4.6.1. Influence of Lower-Order Terms 186

4.6.2. Higher-Order Forms 187

4.6.3. Predictor-Corrector Methods 190

4.6.4. Asymmetric Approximations 192

4.6.5. Variable Coefficients 199

4.6.6. Nonlinear Parabolic PDEs 203

4.6.7. The Box Method 211

4.7. Solution of Finite Difference Approximations 213

4.7.1. Solution of Implicit Approximations 214

4.7.2. Explicit versus Implicit Approximations 218

4.8. Composite Solutions 219

4.8.1. Global Extrapolation 220

4.8.2. Some Numerical Results 224

4.8.3. Local Combination 226

4.8.4. Some Numerical Results 230

4.8.5. Composites of Different Approximations 231

4.9. Finite Difference Approximations in Two Space Dimensions 234

4.9.1. Explicit Methods 234

4.9.2. Irregular Boundaries 240

4.9.3. Implicit Methods 241

4.9.4. Alternating Direction Explicit (ADE) Methods 244

4.9.5. Alternating Direction Implicit (ADI) Methods 245

4.9.6. LOD and Fractional Splitting Methods 255

4.9.7. Hopscotch Methods 261

4.9.8. Mesh Refinement 264

4.10. Three-Dimensional Problems 265

4.10.1. ADI Methods 266

4.10.2. LOD and Fractional Splitting Methods 272

4.10.3. Iterative Solutions 274

4.11. Finite Element Solution of Parabolic Partial Differential Equations 276

4.11.1. Galerkin Approximation to the Model Parabolic Partial Differential Equation 277

4.11.2. Approximation of the Time Derivative 280

4.11.3. Approximation of the Time Derivative for Weakly Nonlinear Equations 282

4.12. Finite Element Approximations in One Space Dimension 285

4.12.1. Formulation of the Galerkin Approximating Equations 285

4.12.2. Linear Basis Function Approximation 289

4.12.3. Higher-Degree Polynomial Basis Function Approximation 294

4.12.4. Formulation Using the Dirac Delta Function 297

4.12.5. Orthogonal Collocation Formulation 299

4.12.6. Asymmetric Weighting Functions 306

4.13. Finite Element Approximations in Two Space Dimensions 309

4.13.1. Galerkin Approximation in Space and Time 309

4.13.2. Galerkin Approximation in Space Finite Difference in Time 314

4.13.3. Asymmetric Weighting Functions in Two Space Dimensions 316

4.13.4. Lumped and Consistent Time Matrices 321

4.13.5. Collocation Finite Element Formulation 330

4.13.6. Treatment of Sources and Sinks 339

4.13.7. Alternating Direction Formulation 342

4.14. Finite Element Approximations in Three Space Dimensions 348

Chapter 5. Elliptic Partial Differential Equations 355

5.1. Model Elliptic PDEs 355

5.1.1. Specific Elliptic PDEs 355

5.1.2. Boundary Conditions 356

5.1.3. Further Items 358

5.2. Finite Difference Solutions in Two Space Dimensions 360

5.2.1. Five-Point Approximations and Truncation Error 360

5.2.2. Nine-Point Approximations and Truncation Error 371

5.2.3. Approximations to the Biharmonic Equation 373

5.2.4. Boundary Condition Approximations 375

5.2.5. Matrix Form of Finite Difference Equations 377

5.2.6. Direct Methods of Solution 383

5.2.7. Iterative Concepts 385

5.2.8. Formulation of Point Iterative Methods 392

5.2.9. Convergence of Point Iterative Methods 405

5.2.10. Line and Block Iteration Methods 418

5.2.11. ADI Methods 421

5.2.12. Acceleration and Semi-Iterative Overlays 430

5.3. Finite Difference Solutions in Three Space Dimensions 434

5.3.1. Finite Difference Approximations 435

5.3.2. Iteration Concepts 437

5.3.3. ADI Methods 437

5.4. Finite Element Methods for Two Space Dimensions 441

5.4.1. Galerkin Approximation 442

5.4.3. Collocation Approximation 449

5.4.4. Mixed Finite Element Approximation 453

5.4.5. Approximation of the Biharmonic Equation 455

5.5. Boundary Integral Equation Methods 461

5.5.1. Fundamental Theory 461

5.5.2. Boundary Element Formulation 465

5.5.4. Linear Interpolation Functions 471

5.5.5. Poisson's Equation 473

5.5.6. Nonhomogeneous Materials 475

5.5.7. Combination of Finite Element and Boundary Integral Equation Methods 478

5.6. Three-Dimensional Finite Element Simulation 481

Chapter 6. Hyperbolic Partial Differential Equations 486

6.1. Equations of Hyperbolic Type 487

6.2. Finite Difference Solution of First-Order Scalar Hyperbolic Partial Differential Equations 489

6.2.1. Stability, Truncation Error, and Other Features 490

6.2.2. Other Approximations 497

6.2.3. Dissipation and Dispersion 505

6.2.4. Hopscotch Methods and Mesh Refinement 524

6.3. Finite Difference Solution of First-Order Vector Hyperbolic Partial Differential Equations 526

6.4. Finite Difference Solution of First-Order Vector Conservative Hyperbolic Partial Differential Equations 528

6.5. Finite Difference Solutions to Two- and Three-Dimensional Hyperbolic Partial Differential Equations 539

6.5.1. Finite Difference Schemes 540

6.5.2. Two-Step, ADI, and Strang-Type Algorithms 545

6.5.3. Conservative Hyperbolic Partial Differential Equations 554

6.6. Finite Difference Solution of Second-Order Model Hyperbolic Partial Differential Equations 562

6.6.1. One-Space-Dimension Hyperbolic Partial Differential Equation 562

6.6.2. Explicit Algorithms 564

6.6.3. Implicit Algorithms 569

6.6.4. Simultaneous First-Order Partial Differential Equations 571

6.6.5. Mixed Systems 577

6.6.6. Two- and Three-Space-Dimensional Hyperbolic Partial Differential Equations 580

6.6.7. Implicit ADI and LOD Methods 583

6.7. Finite Element Solution of First-Order Model Hyperbolic Partial Differential Equations 589

6.7.1. Galerkin Approximation 589

6.7.2. Asymmetric Weighting Function Approximation 594

6.7.3. An H[superscript -1] Galerkin Approximation 598

6.7.4. Orthogonal Collocation Formulation 599

6.7.5. Orthogonal Collocation with Asymmetric Bases 604

6.7.6. Dissipation and Dispersion 605

6.8. Finite Element Solution of Two- and Three-Space-Dimensional First-Order Hyperbolic Partial Differential Equations 620

6.8.1. Galerkin Finite Element Formulation 620

6.8.2. Orthogonal Collocation Formulation 622

6.9. Finite Element Solution of First-Order Vector Hyperbolic Partial Differential Equations 625

6.9.1. Galerkin Finite Element Formulation 626

6.9.2. Dissipation and Dispersion 627

6.10. Finite Element Solution of Two- and Three-Space-Dimensional First-Order Vector Hyperbolic Partial Differential Equations 645

6.10.1. Galerkin Finite Element Formulation 645

6.10.2. Boundary Conditions 648

6.11. Finite Element Solution of One-Space-Dimensional Second-Order Hyperbolic Partial Differential Equations 655

6.11.1. Galerkin Finite Element Formulation 655

6.11.2. Time Approximations 657

6.11.3. Dissipation and Dispersion 663

6.12. Finite Element Solution of Two- and Three-Space-Dimensional Second-Order Hyperbolic Partial Differential Equations 665

6.12.1. Galerkin Finite Element Formulation 665.

Notes:

"A Wiley-Interscience publication."

Includes bibliographies and index.

ISBN:

0471098663

OCLC:

7837382