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

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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