Intro -- Preface -- Contents -- About the Authors -- Part I Finite Difference Method -- 1 Difference Scheme -- 1.1 Taylor Series Expansion -- 1.2 Method of Polynomial Interpolation -- 1.3 Being-Determined Coefficient Method -- 1.4 Integral Methods -- 1.5 Method of Characteristics -- 1.6 Control Volume Method -- 1.7 Difference Operators -- 2 The Compatibility, Convergence, and Stability of Difference Schemes -- 2.1 Theoretical Basis of Finite Difference Methods -- 2.2 Compatibility and Truncation Error of Difference Scheme -- 2.3 Convergence and Discretization Errors -- 2.4 Stability and Round-Off Error -- 2.5 Von-neumann Method -- 2.6 A New Method to Get a Transition Matrix from Difference Operator Transformation -- 2.7 Further Discussion of Lax Equivalence Theorem, Convergence, and Stability -- 2.8 Other Methods for Stability Analysis -- References -- 3 Common Difference Schemes for Several Model Equations -- 3.1 Convection Equation -- 3.2 Diffusion Equation -- 3.3 Convection Diffusion Equation -- 4 Difference Schemes for Multi-dimensional Problems -- 4.1 Explicit and Implicit Schemes for Multi-dimensional Problems -- 4.2 Alternating Direction Method -- 4.3 Operator-Splitting Method -- 4.4 Predictor-Corrector Scheme -- 5 |