Numerical Solution of Differential Equations: Introduction to Finite Difference and Finite Element Methods

Contents......Page 3
Preface......Page 6
Intro......Page 7
--- Finite Difference Methods......Page 12
A Simple Example of a Finite Difference Method......Page 13
Fundamentals of Finite Difference Methods......Page 18
Deriving FD Formulas Using the Method of Undetermined Coefficient......Page 23
Consistency, Stability, Convergence, and Error Estimates of FD Methods......Page 25
FD Methods for 1D Self-adjoint BVPs......Page 31
FD Methods for General 1D BVPs......Page 33
The Ghost Point Method for Boundary Conditions Involving Derivatives......Page 34
An Example of a Nonlinear BVP......Page 38
The Grid RefinementAnalysis Technique......Page 41
1D IIM for Discontinuous Coefficient......Page 43
Exercises......Page 48
FDMs for 2D Elliptic PDEs......Page 51
Boundary and Compatibility Conditions......Page 53
The Central Finite Difference Method for Poisson Equations......Page 55
The Maximum Principle and Error Analysis......Page 59
Finite Difference Methods for General Second-order Elliptic PDEs......Page 64
Solving the Resulting Linear System of Algebraic Equations......Page 65
A Fourth-Order Compact FD Scheme for Poisson Equations......Page 71
A Finite Difference Method for Poisson Equations in Polar Coordinates......Page 73
Programming of 2D Finite Difference Methods......Page 76
Exercises......Page 79
FDMs for Parabolic PDEs......Page 82
The Euler Methods......Page 84
The Method of Lines......Page 89
The Crank–Nicolson Scheme......Page 91
Stability Analysis for Time-dependent Problems......Page 93
FD Methods and Analysis for 2D Parabolic Equations......Page 101
The ADI Method......Page 103
An Implicit–explicit Method for Diffusion and Advection Equations......Page 108
Exercises......Page 109
FDMs for Hyperbolic PDEs......Page 112
Characteristics and Boundary Conditions......Page 113
Finite Difference Schemes......Page 114
The ModifiedPDE and Numerical Diffusion/Dispersion......Page 119
The Lax–Wendroff Scheme and Other FD methods......Page 121
Numerical Boundary Conditions......Page 124
Finite Difference Methods for Second-order Linear Hyperbolic PDEs......Page 125
Finite Difference Methods for Conservation Laws......Page 131
Exercises......Page 135
--- Finite Element Methods......Page 137
The Galerkin FE Method for the 1D Model......Page 138
Different Mathematical Formulations for the 1D Model......Page 141
Key Components of the FE Method for the 1D Model......Page 146
Matlab Programming of the FE Method for the 1D Model Problem......Page 155
Exercises......Page 159
Functional Spaces......Page 161
Spaces for Integral Forms,......Page 163
Sobolev Spaces and Weak Derivatives......Page 167
FE Analysis for 1D BVPs......Page 171
Error Analysis of the FE Method......Page 176
Exercises......Page 181
Boundary Conditions......Page 184
The FE Method for Sturm–Liouville Problems......Page 188
High-order Elements......Page 192
1D Matlab FE Package......Page 198
The FE Method for Fourth-Order BVPs in 1D......Page 211
The Lax–Milgram Lemma and the Existence of FE Solutions......Page 217
1D IFEM for Discontinuous Coefficient......Page 224
Exercises......Page 226
The Second Green’s Theorem and Integration by Parts in 2D......Page 231
Weak Form of Second-Order Self-Adjoint Elliptic PDEs......Page 234
Triangulation and Basis Functions......Page 236
Transforms, Shape Functions, and Quadrature Formulas......Page 249
Some Implementation Details......Page 251
Simplificationof the FE Method for Poisson Equations......Page 254
Some FE Spaces in......Page 260
The FE Method for Parabolic Problems......Page 275
Exercises......Page 278
General Procedure......Page 280
A.1 System of First-Order ODEs of IVPs......Page 281
A.2 Well-posedness of an IVP......Page 282
A.3 Some Finite Difference Methods for Solving IVPs......Page 283
A.4 Solving IVPs Using Matlab ODE Suite......Page 286
Exercises......Page 290
Refs......Page 291
Index......Page 293